There exists a strategy by which B can be sure to win if and only if A is of first category. Let f1' f2' Let 10 denote the interior of any interval I. Let S be a sequence consisting of all closed intervals that have rational endpoints and are contained in Ig. Let J 1 be the first term of S. It is easy to verify, using 1 , that this construction defines inductively a sequence Ji having the required properties.
Now consider an arbitrary sequence of positive integers in' and define 7 From 4 and 5 it follows that conditions 2 and 3 are satisfied for all n; hence the nested sequence In is a possible play ofthe game consistent with the given strategy for B. By hypothesis, the set n In must be contained in B. If this sequence is used to define 7 , then x E E C B. Consequently, n in E there is a unique sequence i 1 , i 2 , This shows that. There exists a strategy by which A can be sure to win if and only if 11 nB is of first category for some interval 11 C 10 ,.
If such an interval exists, A can start by choosing it' for 11 , Then, by an obvious strategy, he can insure that n In is disjoint to B. Since the intersection is non-empty, this is a winning strategy for A. On the other hand, if A has a winning strategy he can always modify it so as to insure that the intersection of the intervals In will consist of just one point of A. By Theorem 6. If the set A has the property of Baire, then B or A possesses a winning strategy according as A is of first or second category. If G is empty, then B has a winning strategy, by Theorem 6.
If G is not empty, A has only to choose 11 C G to insure that he will be able to win. Otherwise, E is said to be of second category at the point x. These notions are analogous to the metric notion of density discussed in Chapter 3. The set G of points at which A' is of first category is open. Hence G is the same as the regular open set that appears in Theorem 4. The fact that G differs from A by a set of first category is analogous to the Lebesgue density theorem.
By Theorems 6. Is it possible that neither may hold? Let A be the intersection of 10 with a Bernstein set. Then neither A nor B contains an uncountable Go set Lemma 5. For if one is of first category, the other is a set of second category having the property. Consequently, this game A, B is not determined in favor of either player. The possibility of indeterminateness makes the Banach-Mazur game particulary interesting for the general theory of games.
It also raises some interesting questions. If a game is determined in favor of one of the players, should it be called a game of "skilr'? If neither player can control the outcome, is the outcome a matter of "chance"? What does "chance" mean in this connection?
There is another version of the Banach-Mazur game, in which the players alternately choose successive blocks of digits of arbitrary finite length in the decimal or binary development of a number. If the number so defined belongs to A, A wins; otherwise, B wins. In effect, this is the same as the game with intervals, except that now all the intervals are required to be decimal intervals.
Any winning strategy for the original game can easily be modified so as to satisfy this condition, and Theorems 6. However, if the blocks are all required to be of length 1, that is, if A and B alternately choose successive digits in the development of a number, then we have an entirely different game, a game which was first studied by Gale and Stewart . The conditions under which one of the players now has a winning strategy are still not completely understood.
It is not known, for instance, whether this game is determined in favor of one or the other player whenever A is a Borel set. Recent results suggest that the answer may depend upon what set-theoretic axioms one assumes [22, p. Let f be a real-valued function on R. The set D of all points at which f is dis- continuous can be represented in the form. Thus Theorem 7.
For any F" set E there exists a bounded function f having E for its set of points of discontinuity. Such a function need not be continuous, as simple examples show. However, the following theorem shows that a function of first class cannot be everywhere discontinuous. It is known as Baire's theorem on functions of first class. More exactly, it is a part of Baire's theorem. It was in this connection that Baire originally introduced the notion of category. Theorem 7. If f can be represented as the limit of an everywhere convergent sequence of continuous functions, then f is continuous except at a set of points of first category.
This should be compared with the well-known theorem that the limit of a uniformly convergent sequence of continuous functions is everywhere continuous. Consider any closed interval I. Hence, for some positive integer n, En n I contains an open interval J. This shows that F is nowhere dense. With only slight changes in wording, it applies when f and all of the functions fn are restricted to an arbitrary perfect set P. In this case the notion of category must be interpreted relative to P. The Baire category theorem remains true: if an open interval I meets P, then no countable union of sets nowhere dense relative to P can be equal to In P.
Thus if f is any func-. Conversely, Baire showed that any such function is of first class. For an elementary proof, see [4, Note II]. We shall not prove this, but merely note that a simple example shows that the converse of Theorem 7. But the restriction of f to C is discontinuous at every point of C, hence f is not of first class. It is easy enough to formulate a necessary and sufficient condition for the conclusion of Theorem 7. The set of points of discontinuity of f is of first category if and only if f is continuous at a dense set of points.
This is an immediate consequence of Theorem 7. To illustrate how it serves to answer several natural questions, we mention two examples. It is well known that a trigonometric series may converge pointwise to a discontinuous function. How discontinuous can the sum function be? Can the sum of an everywhere convergent trigonometric series be everywhere discontinuous? Again, it is well known that the derivative of an everywhere differen- tiable function f need not be everywhere continuous.
Can the derivative of an everywhere differentiable function be everywhere discontinuous? Having found conditions under which the set D of points of dis- continuity of a function is of first category, it is natural to inquire under what conditions D is a nullset. One answer is provided by the following well-known Theorem 7. In order that a function f be Riemann-integrable on every finite interval it is necessary and sufficient that f be bounded on every finite interval and that its set of points of discontinuity be a nullset.
Suppose the contrary. These intersect in a point x of I. Choose n so that In C J. Any continuous function on a closed interval is integrable. It may be noted that the above proof of this fact did not involve the notion of uniform continuity. Now, to prove Theorem 7. Then for any positive integer k, I can be divided into intervals 11' Since Fk is a bounded closed nullset, it is possible to cover Fk with a finite number of disjoint open intervals the sum of whose lengths is less than 11k. Hence, by Lemma 7.
Consequently, f is Riemann-integrable on I. To round out this discussion of points of discontinuity, one may ask whether there is a natural class of functions that is characterized by having only countably many discontinuities. One answer is provided by Theorem 7.
The set of points of discontinuity of any monotone function f is countable. Any countable set is the set of points of dis- continuity of some monotone function. Hence the set of points of discontinuity of f is countable. The function f x. This should be compared with the much deeper theorem, due to Lebesgue, that any monotone function is differentiable has a finite derivative except at a set of points of measure zero [31, p.
The Theorems of Lusin and Egoioff. A real-valued function f on R is called measurable if f - 1 U is measurable for every open set U in R. In either definition, U may be restricted to some base, or allowed to run over all Borel sets. Q, where G is open, F is closed, and P and Q are of first category. More generally, continuity and the property of Baire are related as follows [18, p. Theorem 8. A real-valued function f on R has the property of Baire if and only if there exists a set P of first category such that the restriction of f to R - P is continuous.
Pj, where Gj is open and Pj is of first category. Q for some set QC P. Thus f has the property of Baire. If 9 denotes the restriction of f to R - E, then. Hence g-I Uj is both closed and open relative to R - E, and it follows that 9 is continuous. To see this we construct an example as follows. Since A and R - A have positive measure in every interval, the restriction of f to the complement of any nullset is nowhere continuous.
The following result, known as EgorojJ's theorem, establishes a relation between convergence and uniform convergence. For any two positive integers nand k let. Thus In converges to I uniformly on E - F. This is shown by the following example. This shows that In does not converge uniformly on a, b.
Let E be any set on which In does converge uniformly. Hence In converges to 0 uniformly on E. From what we have shown, E cannot contain an interval. The usefulness of the notion of category only becomes fully apparent in more general spaces, especially metric spaces. Let us recall the basic definitions. This notion due to Frechet is a natural abstraction of some of the properties of distance in a Euclidean space of any number of dimensions. Many theorems in analysis become simpler and more intuitive when formulated in terms of a suitable metric. A sequence is convergent if it converges to some point of X.
Balls are open sets, and arbitrary unions and finite intersections of open sets are open. Any class. A subclass. The open subsets of any metric space X constitute a topology in X, but not every topology can be represented in this way. A metric space is called separable if it has a countable dense subset, or, equivalently, a countable base. Two metrics in a set X are topologically equivalent if they determine the same topology. Very often it is the topological structure of a metric space that is of primary interest, and the. Any property that is definable in terms of open sets alone is a topological property.
The complement of an open set is called closed. The smallest closed set that contains a set A is called the closure of A; it is denoted by A or A -. Similarly, the largest open set contained in A is called the interior of A; it is equal to A' -'. A is a neighborhood of x if x belongs to the interior of A. A set A is nowhere dense if the interior of its closure is empty, that is, if for every non-empty open set G there is a non-empty open set H contained in G - A.
A set is of first category if it can be represented as a countable union of nowhere dense sets; otherwise, it is of second category. Fa sets, Go sets, Borel sets, and sets having the property of Baire are defined exactly as before. All of these are topological properties of sets, and the definitions apply to any topological space. A mapping f of a topological space X into a topological space Y is continuous at the point Xo in X iffor every open set V that contains f xo there is a neighborhood U of Xo such that f x E V for every x E U.
A one-to-one mapping f of X onto Y is called a homeomorphism if both f and f - 1 are continuous. When such a mapping exists, X and Y are said to be homeomorphic or topologically equivalent. Two metrics e and a in a set X are topologically equivalent if and only if the identity mapping of X onto itself is a homeomorphism of X, e onto X, a.
Every convergent sequence is Cauchy, but the converse is not generally true. However, there is an important class of spaces in which every Cauchy sequence is convergent. Such a metric space is said to be complete. For instance, the real line is complete with respect to the usual metric Ix - YI. Here X is complete but Y is not. If f is a homeomorphism of X, 0 onto a com- plete space Y,O" , then O" j x , fey is a metric in X topologically equivalent to o. Thus a metric space is topologically complete if and only if it can be remetrized with a topologically equivalent metric so as to be complete.
An important property of such spaces is that the Baire category theorem still holds. Theorem 9. This can be done step by step, taking for Sn a ball with center Xn in Sn- 1 - An which is non-empty because An is nowhere dense and with sufficiently small radius. This shows that X - A is dense in X.
A topological space X is called a Baire space if every non-empty open set in X is of second category, or equivalently, if the complement of every set of first category is dense. In a Baire space, the complement of any set of first category is called a residual set. Then B is a G6 set contained in E. Since X is a Baire space, it follows that B is dense in X.
Examples of Metric Spaces. Let C, or C[a, b], denote the set of all real-valued continuous functions f on the interval [a, b], and define! It is easy to verify that! Convergence in this metric means uniform convergence on [a, b]. For this reason,! It therefore converges to a limit f x. Thus fi converges to f uniformly on [a, b]. By a well-known theorem, it follows that f is con- tinuous on [a,b]. This shows that the space C,! Again it is easy to verify that all the axioms are satisfied. To see that this metric is not topologically equivalent to!
Hence fn is a Cauchy sequence. Next consider the set R[a, bJ of Riemann-integrable functions on [a, bJ, with the same metric a. Here we encounter a difficulty: the fourth axiom is not satisfied. A set X with a distance function that satisfies only the first three axioms is called a pseudo metric space. It is easy to verify that this defines an equivalence relation in X, and that the value of O x, y depends only on the equivalence classes to which x and y belong.
If we take these classes as the elements of a set X, then X, 0 is a metric space. Let 1 denote the equivalence class to which f belongs. Thus R, a contains a subset isometric to C, a. Hence no element of the c-neighborhood of 9 belongs to EM. Since 9 can be taken arbitrarily close to 10' this shows that EM is nowhere dense in R. Hence R is of first category in itself. It follows that R is not a complete space. Moreover, no remetrization can make it complete, since category is a topological property.
The fourth axiom holds when we identify sets that differ by a nullset. Thereby we obtain a metric space S, 1. To show that this space is complete, let En be any Cauchy sequence in S, 1. All of these sets belong to S. It follows that En converges to Ein S, 1. This is hardly the easiest way to introduce Lebesgue integration, but it provides one motivation for enlarging the class of integrable functions.
Since R, a is of first category in itself, it is of first category in any space that contains it topologically. In particular, R is of first category in the space of Lebesgue integrable functions . Many examples of nowhere differentiable continuous functions are known, the first having been constructed by Weierstrass. One of the simplest existence proofs is due to Banach [18, p.
It is based on the category method. Banach showed that, in the sense of category, almost all continuous functions are nowhere differentiable; in fact, it is exceptional for a continuous function to have a finite one-sided derivative, or even to have bounded difference quotients on either side, anywhere in an interval. Therefore f belongs to En. Then at each point of [0,1 the function h has a one-sided derivative on the right numerically greater than n. Therefore hE C - En. This is the set of all continuous functions that have bounded right difference quotients at some point of [0, 1.
Similarly, the set offunctions that have bounded left difference quotients at some point of 0,1] is of first category; indeed, this can be deduced from what we have already shown by considering the isometry of C induced by the substitution of 1 - x for x. The union of these two sets includes all functions in C that have a finite one-sided derivative somewhere in [0, 1].
By similar reasoning, one can show that a residual set of functions in C have nowhere an infinite two-sided derivative. One may ask whether it is possible to go even further and find a continuous function that has nowhere a finite or infinite one-sided derivative. Such a strongly nowhere differentiable function was first constructed by Besicovitch in However, it is a remarkable fact that the existence of such functions cannot be demonstrated by the category method; Saks showed that the set of such functions is only of first category in C! See [18, p. The use of the Baire category theorem to prove that a set is non- empty amounts to a demonstration of the fact that a member of the set can be defined as the limit of a suitably constructed sequence.
The advantage of the category method is that it furnishes a whole class of examples, not just one, and it generally simplifies the problem, enabling one to concentrate on the essential difficulty.
Lebesgue measure and integration
When it succeeds, an example can always be constructed by successive approxima- tion, starting anywhere in the space. At least, this is true in principle. But if the proof of nowhere denseness is indirect, or long and involved, it may be difficult to obtain an explicit example in this way. Any subset of a metric space is itself a metric space, with the same distance function.
It is obvious that any closed subset of a complete metric space is complete with respect to the same metric. When can a subspace be remetrized so as to be complete? This question is answered by the following Theorem Any non-empty Gil subset of a complete metric space is topologically complete, that is, the subset can be remetrized so as to be complete. Following Kuratowski we base the proof on the following Lemma Then X can be remetrized so as to be complete. Define a new distance function in X by.
To verify the triangle axiom it suffices to observe that it is satisfied by each term. The other axioms are clearly satisfied. Thus a and Q are equivalent metrics. Then for any positive integer i there is an integer N such that. We may assume that each of the sets Fi is non-empty. Then d x, FJ is a real-valued continuous function on Y, positive on X. Then Xn converges in Y to some point y, since Y is complete. Hence y E Gi for every i; that is, y E X. Consequently, by Lemma Theorem Let f be a homeomorphism of X onto Y.
Then Gn is open in Z. Let ZEn Gn. Since Xn. It is an almost trivial observation that there exists a homeomorphism h of I onto itself such that h F is a nullset. This is a strictly increasing continuous map of I onto itself. The intervals that compose G are mapped onto a sequence of intervals of total length 1. Hence h F is a nullset. The generalization of this result to a set of first category cannot be proved in the same way. Nevertheless, the conclusion still holds, as we shall show by a category argument.
The space C l ' Q is complete, since C 1 is a closed subset of C. However, the space H, Q is not complete. Therefore hog belongs to 0. Consequently, 0. If hE E, then h An is a nullset for every n. By Lemma 5. Then h is a strictly increasing continuous map of [0, 1] onto itself. Let f be a bounded function on [0,1], and let D be its set of points of discontinuity. Let h be an arbitrary automorphism of [0, 1]. Then the composite func- tion f h is bounded and has the set h - 1 D for its set of points of dis- 0. We know Theorem 7.
If D is un- countable, it contains an uncountable closed set. Then, for some h, the set h - 1 D has positive measure. On the other hand, if D is countable then h - 1 D is countable and has measure zero for every h. If D is of first category, there exists an h such that h - 1 D is a nullset.
On the other hand, if D is of second category then one of the closed sets of which it is the union must contain an interval. In this case, h -1 D also contains an interval, and is never a null set. Recalling Theorem 7. Let f be a bounded function on [0, 1], and let D be its set of points of discontinuity. Let h be an arbitrary homeomorphism of [0, 1] onto itself. Then the composite function f chis Riemann integrable a for all h if and only if D is countable, b for some h if and only if D is of first category, c for the identity mapping h if and only if D is a nullset.
In this theorem each of the a-ideals we have been considering answers a question concerning the effect of a strictly monotone substitution on the Riemann integrability of a function! Another consequence is the following characterization of sets of first category, in which the notion of a nowhere dense set does not appear. A linear set A is of first category if and only if there exists a homeomorphism h of the line onto itself such that h A is contained in an Fa nullset.
This theorem characterizes sets of first category as those that are topologically equivalent to a special kind of nullset. Any set A of first category is contained in an Fa set B of first category. Divide the line into non-overlapping intervals Ii of unit length.
Let hi be an automorphism of Ii that leaves the endpoints fixed and maps B n Ii onto a nullset. The mappings hi define an automorphism h of the line such that h B is a nullset. Therefore h A is contained in the Fa nullset h B. Conversely, let A be any subset of an Fa nullset. Therefore each of the sets Fn is nowhere dense. Consequently A, and its image under any automorphism of the line, is of first category.
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Fubini's Theorem. Linear Lebesgue measure is defined by covering sequences of intervals, and plane measure by covering sequences of rectangles. We shall now consider how these measures are related to each other. It is clear what kind of answer we should expect. Thus the area is computed "by slicing. We shall not formulate the theorem in full generality, but confine attention to the case in which A is a nullset. Then the theorem asserts that almost all vertical or horizontal sections of A have measure zero.
Let X and Y be two non-empty sets. For instance, in coordinate geometry the plane is repre- sented as the product of two lines. Note that Ex is a subset of Y, not of X x Y. The operation of sectioning commutes with union, intersection, and complement. Lemma A set A has Lebesgue measure zero if and only if it can be covered infinitely many times by a sequence of intervals In such that the series L IInl is convergent. Hence A is a nullset. If E is a plane set of measure zero, then Ex is a linear nullset for all x except a set A of linear measure zero.
Hence, by 2 ,. From 5 , it follows that A is a linear nullset. If E is a plane measurable set, then Ex is linearly measurable for all x except a set of linear measure zero. Any section of a closed set is closed, hence Ax is an Fa for every x. By Fubini's theorem, N x is a nullset for almost all x. Since Ex is measurable for any such x, the conclusion follows. We shall not prove this. The usual proof depends on properties of the Lebesgue integral, which we have not developed here.
We remark only that the conclusion does not follow unless E is assumed to be measurable. This is shown by the following theorem, due to Sierpinski. There exists a plane set E such that a E meets every closed set of positive plane measure, and b no three points of E are collinear.
Such a set E cannot be measurable. For if E were measurable, then a and Theorem 3. Hence E is not measurable, and therefore not a nullset, despite the fact that each of its sections has at most two points. For Sierpinski's proof of the above theorem, see Fund.
We shall give here only a simplified version, assuming the con- tinuum hypothesis. Let the class of closed sets of positive plane measure be well ordered in such a way that each member has only countably many predecessors. Then choose P3 in F3 not collinear with Pl and P2. Fubini's theorem has a category analogue. In its general formulation, this theorem was proved in by Kuratowski and Ulam [18, p.
If E is a plane set of first category, then Ex is a linear set of first category for all x except a set of first category. If E is a nowhere dense subset of the plane X x Y, then Ex is a nowhere dense subset of Y for all x except a set of first category in X. The two statement are essentially equivalent. Hence the first statement follows from the second. If E is nowhere dense, so is E, and Ex is nowhere dense whenever E x is of first category.
Hence the second statement follows from the first. It is therefore sufficient to prove the second statement for any nowhere dense closed set E. Then G is a dense open subset of the plane. Hence Gn contains points of U. Therefore Gn is a dense open. For any x En n subset of X, for each n. Consequently, the set Gn is the complement of Gn , the section Gx contains points of Vn for every n. This shows that for all x except a set of first category, Ex is nowhere dense. In fact, it is sufficient to assume that there is a sequence of non-empty open sets in Y such that every non-empty open set contains a member of the sequence.
P, where G is open and P is of first category. Px , for all x. Every section of an open set is open, hence Ex has the property of Baire whenever Px is of first category. By Theorem A product set A x B is of first category in X x Y if and only if at least one of the sets A or B is of first category. Similar reasoning applies to B. Conversely, if A x B is of first category and A is not, then by Theorem If E is a subset of X x Y that has the property of Baire, and if Ex is of first category for all x except a set of first category, then E is of first category.
P, where P is of first category and G is an open set of second category. This is clear in the case of the plane. In general it follows from the Banach category theorem, which will be discussed in the next chapter. Therefore Ex is of second category for all x in U except a set of first category. This implies that Ex is of second category for all x in a set of second category, contrary to hypothesis.
There exists a plane set E of second category such that no three points of E are collinear. The class of plane Ga sets of second category has power c. Since the set of all lines. The set of all points p" so chosen contains no three collinear points. It is of second category because its complement contains no Gb set of second category. We shall show that, in a sense, the Kuratowski- Ulam theorem can be reduced to Fubini's. The reduction is limited to the case of plane sets, and it does not lead to any simplification.
The interest of the question lies in the technical problem it poses: to find a transforma- tion of the plane that will reduce any given instance of the Kuratowski- Ulam theorem to a case of Fubini's. No similar reduction of Fubini's theorem to that of Kuratowski-Ulam appears possible. In Chapter 13 it was shown that any linear set of first category can be transformed into a nullset by an automorphism of the line.
Similarly, it can be shown that any set of first category in r-space can be transformed into one of measure zero by an automorphism of the space . This was first proved in for subsets ofthe square by L. Brouwer . However, this result is inadequate for our present purpose, because such a transformation need not take sections into sections. We shall establish the existence of such an automor- phism by a category argument similar to the one we gave in the 1-dimen- sional case. Let m 2 and m denote 2-dimensional and linear Lebesgue measure, respectively. As in Chapter 13, let H, e denote the space of automorphisms of the unit interval that leave the endpoints fixed.
Let H2 denote the set of all automorphisms of the unit square of the form f x g, where f and 9 belong to H. H2 may be identified with the Cartesian product H x H. Any remetrization of H determines a corresponding remetriza- tion of HZ. Let F be a nowhere dense closed subset of the square. By the same reasoning as in the proof of Theorem In each of the squares I; x Ij , F has a similarly situated hole. Let II be a piece-wise linear automorphism of II' leaving the endpoints fixed, such that. Similarly, let gl be a piece-wise linear automorphism of II' leaving the endpoints fixed, such that.
Similarly, the transformations gj define a trans- formation g E H. The product transformation I x g maps each square Ii x Ij onto itself, hence its cr-distance from the identity is less than e. We have shown that Eik is a dense open subset of the topologically complete space H2. Then I is a homeomor- phism of 0, 1 onto the line. Since f' is continuous and positive, both g and g-1 map nullsets onto nullsets. If E is of first category in the plane, then g - 1 E is a set of first category in the square. Then go hog - 1 is a product automorphism of the plane that maps E onto a nullset.
Let E be a nowhere dense closed. Then each section Ex is either nowhere dense or it contains an interval. Thus f A is an Fa set of measure zero. Hence f A , and therefore A, is of first category, by Theorem In a topological space that has a countable base, it is obvious that the union of any family of open sets of first category is of first category. One need only take the union of those members of the base that are contained in at least one member of the given family. The same reasoning shows that the union of any family of open sets of measure zero has measure zero for any measure defined for all open sets.
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It is remarkable that the first statement remains valid whether the space has a countable base or not. The second statement, however, needs to be qualified. In a topological space X, the union of any family of open sets of first category is of first category. Then the closed set G- Uff is nowhere dense. To discuss the analogue of Theorem Then Fa.
Evidently any cardinal less than one of measure zero has measure zero. As already mentioned in Chapter 5, it is known that every cardinal less than the first weakly inaccessible cardinal has measure zero, and that assuming the continuum hypothesis only exceedingly large cardinals can fail to have measure zero. A measure J1 defined on the class of Borel subsets of a space X is called a Borel measure.
Let J1 be a finite Borel measure in a metric space X. For any set E C A we have. Hence the set function. It is evidently a finite measure, and non- atomic. If X is a metric space with a base whose cardinal has measure zero, and if J1 is a finite Borel measure in X, then the union of any family of open sets of measure zero has measure zero.
Measure and Category, John C. Oxtoby
Let f! Kemperman and Maharam  have shown that in the Cartesian product X of c copies of the line it is possible to define a normalized measure Jl on the a-algebra generated by the elementary open sets, with the property that X can be covered by a family of open sets of measure zero. The elementary open sets constitute a base of cardinality c, and c has measure zero assuming the continuum hypothesis by Vlam's Theorem 5.
The following theorem is perhaps the ultimate generalization of Theorem 1. Let X be a metric space with a base whose cardinal has measure zero. Let Jl be a nonatomic Borel measure in X such that i every set of infinite measure has a subset with positive finite measure, and ii every set of measure zero is contained in a Go set of measure zero.
Then X can be represented as the union of a Go set of measure zero and a set of first category. By selecting a point from each member of the given base, we obtain a dense set S of at most the same cardinality. Hence Jl is defined for all subsets of D. Because Jl is zero for points and. The complement of E is a set of first category.
Then all subsets of X are open and a nontrivial finite measure defined for all subsets of X and zero for points would be a Borel measure that satisfies conditions i and ii but not the conclusion. It is easy to verify that any finite Borel measure in a metric space satisfies conditions i and ii. The class of Borel sets that have an F" subset and a Go superset of equal measure is a a-algebra that includes all closed sets. However, these conditions cannot be omitted from Theo- rem In the course of his studies in celestial mechanics, Poincare discovered a theorem which is remarkable both for its simplicity and for its far- reaching consequences.
It is noteworthy also for having initiated the modern study of measure-preserving transformations, known as ergodic theory. From our point of view, this "recurrence theorem" has a special interest, because in proving it Poincare anticipated the notions of both measure and category. Publication of his treatise, "Les methodes nouvelles de la mecanique celeste" , antedated slightly the introduction of either notion. Let X be a bounded open region of r-space, and let T be a homeomor- phism of X onto itself that preserves volume; that is, G and T G have equal volume, for every open set G C X.
Under iteration of T, each point x generates a sequence x, Tx, T2 x, A point x of an open set G is said to be recurrent with respect to G if r x belongs to G for infinitely many positive integers i. In effect, Poincare proved two theorems, which may be stated together as follows. For any open set G C X, all points of G are recurrent with respect to G except a set of first category and measure zero.
The category assertion has to be read between the lines of Poincare's discussion. He began by showing that recurrent points are dense in G. His proof involved the construction of a nested sequence of regions; it may be interpreted as amounting to a proof of Baire's theorem for the case in hand. Since it is a trivial matter to show that the set of points recurrent with respect to G is a Go set, the category assertion may pro- perly be ascribed to Poincare even though he makes no such explicit statement. This part of Poincare's reasoning was subsequently generalized and extended by G.
Birkhoff [3, Chapter 7]. The category assertion was made explicit by Hilmy . The measure assertion of Theorem However, when read against an. It was reformulated in modern terms by Caratheodory . Closer analysis of Poincare's reasoning reveals that the assumed preservation of volume is not really essential. In the first part of his reasoning it is used only to exclude the possibility of an open set whose images are mutually disjoint, and in the second part it serves only to exclude the possibility of such a set having positive measure.
Moreover, there is no need to assume that T is one-to-one. When stripped of inessential features, both parts of Poincare's theorem are seen to be contained in a single abstract recurrence theorem, which we shall now formulate and prove. T is called dissipative if there exists a wandering set that belongs to S - I; otherwise T is called nondissipative. Suppose T is nondissipative. Conversely, if T is dissipative there exists a wandering set E that belongs to S - I.
This shows that T lacks the recurrence property. Both parts of Theorem Suppose first that T is a one-to-one measure-preserving transformation of a bounded open region X of r-space onto itself. Take S to be the cr-algebra of measurable subsets of X, and I to be the cr-ideal of null sets. Then T is. Since the measure of X is finite, any measurable wandering set must be a nullset, therefore T is nondissipative.
Consequently, T has the recurrence property. This means that almost all points of any measurable set E return to E infinitely often under iteration of T. In particular, for any open set G C X, all points of G except a set of measure zero are recurrent with respect to G. Next suppose that T is a homeomorphism of a metric space X onto itself, with the property that there is no non-empty open wandering set. This will be the case if X is a bounded open subset of r-space and T is volume-preserving. Then T is S-measurable. By the Banach category theorem, there is a largest open set H of first category.
Then any non- empty open subset of Y is of second category. Evidently H, and therefore Y, is invariant under T. Let E be any wandering set having the property of Baire. We may assume that G C Y. Consequently G is an open wandering set, and therefore empty. This shows that any wandering set that has the property of Baire is of first category. Thus T is nondissipative, and so it has the recurrence property. This means that if E has the property of Baire, then all points of E except a set of first category return to E infinitely often under iteration of T. In particular, for any open set G C X, all points of G except a set of first category are recurrent with respect to G.
First we need another definition. A point x is said to be recurrent under T if is recurrent with respect to every neighborhood of itself. Poincare called such a point "stable a la Poisson:". If T is a measure- preserving homeomorphism of a bounded open region X of r-space onto itself, then all points of X except a set of first category and measure zero are recurrent under T.
Let Ek be the set of points x in Uk such that Ti x E Uk for at most a finite number of positive integers i. Therefore each point of X - E is recurrent under T.
In classical mechanics, the con- figuration of a system is described by a finite set of coordinates q l' Q2' A "state" of the system is specified by the instantaneous values of these coordinates and of the corresponding momenta Pl,P2'. These points constitute the phase space of the system. The points of the phase space represent all possible states of the system. As the state of the system changes in time, in accordance with the equations of motion that govern the system, the representative point describes a path in the phase space.
If we follow the motion for unit time, any initial point x in the phase space moves to a point Tx. Thus the equations of motion, followed for unit time, determine a transformation T of phase space into itself. The fundamental existence and uniqueness theorems for solutions of systems of differential equations imply that T is a homeomorphism, provided the terms of the equations are sufficiently continuous and differentiable.
Moreover, the Newtonian equations, written in terms of suitable coordinates and momenta Hamiltonian form , are such that the transformation T preserves 2N-dimensional measure. This result is known as Liouville's theorem the same Liouville whom we encountered in Chapter 2. For a conservative system, the total energy is constant. Therefore T transforms any surface of constant energy into itself. For some systems, it can be shown that the part of phase space where the energy is suitably restricted is a bounded open region of 2N-space.
Then Theorem Poisson had attempted to establish this kind of stability in the "restricted problem of three bodies" by an inconclusive argument based on the kind of terms that can appear in certain series expansions. Poincare established the conclusion rigorously and by a revolutionary new kind of reasoning. This was one of the first triumphs of the modern "qualitative"" theory of differential equations, a theory which Poincare initiated.
Subsequent work on the theory of measure-preserving transforma- tions has shown that Poincare's theorem can be vastly improved. The "ergodic theorem" of G. Birkhoff asserts that under a measure- preserving transformation of a set of finite measure onto itself, not only do almost all points of any measurable set E return to E infinitely often, but they return with a well-defined positive limiting frequency.
Evidently this result goes far beyond Theorem Curiously, though, this refinement of Poincare's theorem turns out to be generally false in the sense of category; the set of points where f x is defined may be only of first category. The analogy between category and measure goes a long way here, but eventually it breaks down. Transitive Transformations. We have given many illustrations of the category method, but in most cases it has only served to give a new, sometimes simpler, existence proof for objects whose existence was already known.
Liouville numbers, nowhere differentiable continuous functions, Brouwer's transformation of the square, were known before the category method was applied. It may therefore be of interest to consider one problem whose solution was first obtained by the category method. Problem To find a homeomorphism T of the closed unit square onto itself such that the positive semi orbit x, Tx, T2 X, When X is a complete separable metric space without isolated points, the existence of such a point implies that points whose positive semiorbit is dense constitute a residual set in X. Then x E E if and only if the positive semi orbit of x is dense in X.
Hence Gj is a dense open set and E is residual in X. Thus an equivalent statement of Problem Some spaces admit a transitive automorphism and some do not. For example, no automorphism of the unit interval is transitive; but multiplication by e2 xia, IX irrational, defines a transitive rotation of the unit circle in the complex plane. An explicit example of a transitive automorphism of the plane was given by Besicovitch [2J, but it is not. The existence of such trans- formations was first established by the category method . The category method has also been successfully used to establish the general existence of automorphisms possessing a much stronger property ofthe same kind, called metrical transitivity .
Suppose that T is transitive and that x is a point whose positive semiorbit is dense.
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Choose an open disk V with center x and radius less than e such that T" x is interior to V and Tx, T2 x, Let S be an automorphism of X equal to the identity outside V, and inside V equal to a radial contraction that maps T" D onto a subset of the interior of D. Con- 0. This shows that the transitive automorphisms constitute only a nowhere dense subset of H. Applied to H, Baire"s theorem gives no assurance that such transforma- tions exist. The category method appears to have failed! But suppose we make the problem harder and demand in addition that T preserve measure!
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error introduced by the use of such an approximation.
The polynomial formed by taking some initial terms of the Taylor serie. Lebesgue's father was a typesetter and his mother was a school teacher. His parents assembled at home a library that the young Henri was able to use. His father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. This page is a list of articles related to set theory. Articles on individual set theory topics Algebra of sets Axiom of choice Axiom of countable choice Axiom of dependent choice Zorn's lemma Axiom of power set Boolean-valued model Burali-Forti paradox Cantor's back-and-forth method Cantor's diagonal argument Cantor's first uncountability proof Cantor's paradox Cantor's theorem Cantor—Bernstein—Schroeder theorem Cardinal number Aleph number Beth number Hartogs number Cardinality Cartesian product Class set theory Complement set theory Complete Boolean algebra Continuum set theory Suslin's problem Continuum hypothesis Countable set Descriptive set theory Analytic set Analytical hierarchy Borel equivalence relation Infinity-Borel set Lightface analytic game Perfect set property Polish space Prewellordering Projective set Property of Baire Uniformization set theory Universally mea.
In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener. In mathematics, in the field of functional analysis, the Cotlar—Stein almost orthogonality lemma is named after mathematicians Mischa Cotlar and Elias Stein. It may be used to obtain information on the operator norm on an operator, acting from one Hilbert space into another when the operator can be decomposed into almost orthogonal pieces.
A more general version was proved by Elias Stein. This is a list of misnamed theorems in mathematics. It includes theorems and lemmas, corollaries, conjectures, laws, and perhaps even the odd object that are well known in mathematics, but which are not named for the originator. That is, these items on this list illustrate Stigler's law of eponymy which is not, of course, due to Stephen Stigler, who credits Robert K Merton. Benford's law. This was first stated in by Simon Newcomb, and rediscovered in by Frank Benford.
This result concerning the probability that the winner of an election was ahead at each step of ballot counting was first published by W. The statement may have been made first by Isaac Newton in The matter of a proof was taken up by Colin MacLaurin c. Moreover, the inverse of that restriction is a Borel section of f - it is a Borel isomorphism. Fabec 28 June Fundamentals of Infinite Dimensional Representation Theory. CRC Press. Federer, Herbert; Morse, A. This is a list of probability topics, by Wikipedia page.
It overlaps with the alphabetical list of statistical topics. There are also the outline of probability and catalog of articles in probability theory. For distributions, see List of probability distributions. For journals, see list of probability journals. For contributors to the field, see list of mathematical probabilists and list of statisticians. General aspects Probability Randomness, Pseudorandomness, Quasirandomness Randomization, hardware random number generator Random number generation Random sequence Uncertainty Statistical dispersion Observational error Equiprobable Equipossible Average Probability interpretations Markovian Statistical regularity Central tendency Bean machine Relative frequency Frequency probability Maximum likelihood Bayesian probability Principle of indifference Credal set Cox's theorem Principle of maximum entropy Information entropy Urn problems Extractor Aleatoric, aleatoric music.
An eponym is a person real or fictitious from whom something is said to take its name. The word is back-formed from "eponymous", from the Greek "eponymos" meaning "giving name". In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed that is, containing all its limit points and bounded that is, having all its points lie within some fixed distance of each other.
Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways. One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano—Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and boun. In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover.
This property gi. Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory.
The use of second-order arithmetic also allows many techniques from recursion the. For an elementary calculus-based introduction, see Divergent series on Wikiversity. Abel, letter to Holmboe, January , reprinted in volume 2 of his collected papers. In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero.
Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. In measure-theoretic analysis and related branches of mathematics, Lebesgue—Stieltjes integration generalizes Riemann—Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.
The Lebesgue—Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue—Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue—Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. They find common application in probability and stochastic processes, and in certain branches of analysis including potential theory.
In mathematics, smooth functions also called infinitely differentiable functions and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions. The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry.
In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case. The functions below are generally used to build up partitions of unity on differentiable manifolds.
An example function Definition of the function The non-anal. Statement of the theorem Types of weak compactness A set A can be weakly compact in three different ways: Compactness or Heine-Borel compactness : Every open cover of A admits a finite subcover. Sequential compactness: Every sequence from A has a convergent subsequence whose limit is in A. Limit point compactness: Every infinite subset of A has a limit point in A. In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small.
Statement of the lemma The following statement can be found in Levin's book. This is a list of general topology topics, by Wikipedia page. Basic concepts Topological space Topological property Open set, closed set Clopen set Closure topology Boundary topology Dense topology G-delta set, F-sigma set closeness mathematics neighbourhood mathematics Continuity topology Homeomorphism Local homeomorphism Open and closed maps Germ mathematics Base topology , subbase Open cover Covering space Atlas topology Limits Limit point Net topology Filter topology Ultrafilter Topological properties Baire category theorem Nowhere dense Baire space Banach—Mazur game Meagre set Comeagre set Compactness and countability Compact space Relatively compact subspace Heine—Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable s.
Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "A specific event will occur. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Introduction Probability and randomness. Basic probability Related topics: set theory, simple theorems in the algebra of sets Events Events in probability theory Elementary events, sample spaces, Venn diagrams Mutual exclusivity Elementary probability The axioms of prob. In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized by the fundamental theorem of calculus in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration.
For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon—Nikodym derivative, or density, of a measure. In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem.
The covering theorem is credited to the Italian mathematician Giuseppe Vitali. On the top: a collection of balls; the green balls are the disjoint subcollection. On the bottom: the subcollection with three times the radius covers all the balls. Then there exist. During these years he did research on the mathematics of finance theory and actuarial science as well as the probability theory for which he became famous.
Cantelli's later work was all on probability and it is in this field where his name graces the Borel—Cantelli lemma and the Glivenko—Cantelli theorem. In — he contributed to the theory. The lower bound is expressed in terms of the probabilities for pairs of events. It was stated in the form given above by Petrov in, equation 6.
References Chung, K. Transactions of the.
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In probability theory, Kolmogorov's zero—one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. Tail events are defined in terms of infinite sequences of random variables.
The Hewitt—Savage zero—one law is a theorem in probability theory, similar to Kolmogorov's zero—one law and the Borel—Cantelli lemma, that specifies that a certain type of event will either almost surely happen or almost surely not happen. It is sometimes known as the Hewitt—Savage law for symmetric events.
The Hewitt—Savage zero—one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite permutations of the indices, has probability either 0 or 1 a "finite" permutation is one that leaves all but finitely many of the indices fixed. Somewhat more abstractly, de. In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces and indeed much of the general theory still holds in that case the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Borel—Cantelli lemma topic In probability theory, the Borel—Cantelli lemma is a theorem about sequences of events.
Revolvy Brain revolvybrain. Borel measure topic In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets and thus on all Borel sets. Fatou's lemma topic In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions.
Frostman lemma topic In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. List of lemmas topic This following is a list of lemmas or, "lemmata", i. Monotone convergence theorem topic In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences sequences that are increasing or decreasing that are also bounded. Zero—one law topic Look up zero—one law in Wiktionary, the free dictionary.
Topological zero—one law, related to meager sets Folders related to Zero—one law: Set indices on mathematics Revolvy Brain revolvybrain Probability theory Revolvy Brain revolvybrain Set indices Revolvy Brain revolvybrain.