Lineability within Peano Curves, Martingales, and Integral Theory
This paper is devoted to give several improvements of some known facts in lineability approach. In particular, we prove that (i) the set of continuous mappings from the unit interval onto the unit square contains a closed, -semigroupable convex subset, (ii) the set of pointwise convergent martingales with is -lineable, (iii) the set of martingales converging in measure but not almost surely is -lineable, (iv) the set of sequences of independent random variables, with , , and the property that is almost surely convergent, is -lineable, (v) the set of bounded functions for which the assertion of Fubini’s Theorem does not hold is consistent with -lineable (it is not 2-lineable), (vi) the set of unbounded functions for which the assertion of Fubini’s Theorem does not hold (with infinite integral allowed) is -lineable but not -lineable.
We start with a standard introduction to the idea of lineability (similar openings can be found, for example, in our paper with T. Natkaniec (see ) or in the papers of other authors mentioned below). For more than a decade now, many mathematicians have been looking at the largeness of some sets by constructing algebraic structures inside them. This approach is called lineability. A comprehensive description of this concept as well as numerous examples and some general techniques can be found in the book  and references therein. Following R. Aron, A. Bartoszewicz, S. Głąb, V. Gurariy, D. Pérez-García, and J.B. Seoane-Sepúlveda, let us recall the following notion.
Definition 1. Let be a cardinal number. (1)Let be a vector space and We say that is -lineable if contains a -dimensional subspace of .(2)Let be a semigroup and We say that is -semigroupable if contains a -generated subsemigroup of
This paper is devoted to the improvements of some known facts: in semigroupability (Section 2) and lineability in probability theory (Section 3). In Sections 4 and 5 we present some new results in probability and integral theory. Along this paper we shall use standard set theoretical notation and notions. In particular, we identify an ordinal number with the set of all ordinals Cardinal numbers are those ordinals which are not equipotent with any and by we denote the cardinal successor of a cardinal number . We use standard notation, i.e., .
2. Peano Curves and Semigroupability
In the century, inspired by the result that the unit interval is of the same cardinality as the unit square , G. Peano constructed (see ) a continuous onto mapping For topological spaces let us denote the set of all continuous surjections from to by . In , N. Albuquerque proved that for every pair the set is -lineable.
Let In this section we are interested in the set L. Bernal-González, M.C. Calderón-Moreno, and J. Prado-Bassas proved in  that the set is semigroupable; i.e., it contains -generated semigroup (with respect to coordinate-wise multiplication). Our goal is to prove the following.
Theorem 2. There exists a closed convex set such that it is -semigroupable; i.e., contains a -generated semigroup.
Proof. Let us fix a function (its construction is a trivial modification of Peano’s example) and let be a set of cardinality of sequences that converge to and are free generators (see ). Let , where, for , the function is defined by the formula (by affine we mean line segments in the square that continuously connect points with and with for ). Clearly is continuous for every Let be a semigroup generated by ; i.e., We will show that (the closure of the convex hull of ) has the desired properties. It is easy to see that since is a set of free generators, the set is algebraically independent. Hence, is a -generated semigroup.
Moreover, Indeed, let Then there are , and such that , with for Clearly is continuous. It is easy to see that, by the definition of the functions from , there exist such that . To show that , it is enough to prove that Let and let be defined by . Since and , the function is onto and there exist such that and . Moreover, as , there is with and then (for a function and for by , we mean ). Hence,
Finally, let us observe that the limit of any convergent sequence from is still in . Indeed, let be a sequence of elements of that converges uniformly to a function Obviously is continuous. Moreover, for any point and any there is with We may assume (by compactness of ) that is convergent to some point It is easy to see that so Hence, is also surjective. Thanks to this observation, we have proved that is a closed, convex, and -semigroupable subset of
3. Lineability within Martingales Theory
This section is devoted to some lineability problems in the martingales setting. Let us start with a short introduction to the probability and martingales theory. Till the end of the section let be a probability space (i.e., is a nonempty set, is a -algebra of subsets of , and is a probability measure). A function is called a random variable, whenever it is -measurable (i.e., for every Borel ); its expected value is computed as follows: and its variance is as follows: There are different notions of convergence of a sequence of random variables. We recall those that will be needed later.
Definition 3. Let and () be random variables. We say that the sequence (1)is pointwise convergent to , whenever for every ,(2)is convergent almost everywhere to , whenever almost surely, i.e., (3)is convergent in measure to , whenever
It is clear that pointwise convergence implies convergence almost surely and it implies convergence in measure. Moreover, none of these implications can be reversed in general.
Definition 4. Let be a random variable with finite expected value and let be a -algebra. The conditional expectation of with respect to , denoted by , is any -measurable function that satisfies for every
It is known that the conditional expectation of with respect to is defined uniquely up to a set of measure zero. In the sequel the following properties of conditional expectation will be useful (see ).
Remark 5. (1)Let be random variables, let be a -algebra, and assume that is -measurable. If and have finite expected values, then (2)Let be a random variable and are -algebras. Then (3)Let be a random variable and let be a -algebra such that and are independent (i.e., for all and we have , where is the smallest -algebra with respect to which is measurable). Then i.e., it is a constant random variable.
Definition 6. We say that a sequence of -algebras forms a filtration whenever
Let us state the definition of a martingale.
Definition 7. Let be a filtration and be a sequence of random variables. We say that is a martingale adapted to the filtration if for every (1) is -measurable,(2),(3) almost everywhere.
Remark 8. The set of all martingales adapted to a fixed filtration forms a real vector space (with respect to natural addition and scalar multiplication).
In , the authors proved the following theorem:
Theorem 9 (Thm. 1, ). There is a probability space in which the set of pointwise convergent sequences of random variables with is -lineable.
Our goal is to state a more general result than the above in the setting of martingales.
Theorem 10. Let be a probability space such that there is infinitely many pairwise disjoint sets of positive measures. Then there exist a filtration and -dimensional vector space of martingales adapted to the filtration such that for every nonzero we have that is pointwise convergent but .
Proof. Let be a partition of (i.e., is a disjoint union of all ’s with all ’s) such that for all Let , for . Clearly forms a filtration. For let us define in the following way: for Observe that is a martingale adapted to the filtration . Indeed, for any , (1) is an -measurable;(2);(3)as for we have that and , so for every we have that . Hence, Let Then is a vector space of martingales adapted to the filtration . Clearly its dimension equals as is linearly independent. We will show that has the desired properties. Let be a nonzero martingale of the form for some , and . It is easy to see that converges pointwise to the variable defined as follows: for On the other hand, we have the following estimation for every : Since , there is such that for every In particular, for every we have Hence, as required.
In the martingales setting one may find another somehow surprising example, namely a martingale such that it converges to in measure but not almost surely (see ). The next theorem is based on this construction but provides a large structure of such martingales.
Theorem 11. Let be a probability space, where is the -algebra of Borel subsets of and is the product measure (countable product of the Lebesgue measure on ). There exist a filtration and a -dimensional vector space of martingales adapted to the filtration such that every nonzero converges to 0 in measure but no almost surely.
Proof. For and define a random variable by the formula: for . Clearly, for any and any natural numbers the random variables and are independent.
Now, let us define a sequence in the following way: We will show that forms a martingale adapted to the filtration , where For every , we have that (1) is measurable by the definition,(2) as and , for every ,(3)by Remark 5, we have the following equalities (almost sure): Let , which is a vector space of martingales adapted to the filtration . We will show that has the desired properties. Let be a nonzero martingale; i.e., there are , and such that ; that is, for every Observe that, by the definition of , we have that for every hence Therefore and in particular with ; i.e., converges to in measure. To prove the second assertion, let us firstly observe the following.
Claim 1. For every and any with , we haveand the event depends, at most, on the events for and .
Proof. By an easy inductive argument we may prove that so for , By the definition, we have the following: for , Clearly the above events are independent so The second assertion is a direct consequence of the proof.
Let for . By the Claim, we have the following estimation: and (as for ). Moreover, by the Claim, the events are independent so by the Borel–Cantelli LemmaSince , there exists such that for all we have Let and . Then In particular, by , hence does not tend to almost surely. In particular, is -dimensional vector space.
(End of Proof)
4. Kolmogorov’s Theorem
We start this section with recalling a well-known theorem due to Kolmogorov (see ).
Theorem 12 (Kolmogorov’s theorem, ). Let be a probability space and be a sequence of independent random variables such that for all and Then, for the sequence is convergent almost surely.
Examples of sequences of independent random variables with expected values equal to and divergent series of their variations such that does converge almost surely are known (see ). Here we show that there is a -dimensional vector space of such sequences.
Theorem 13. There exist a probability space and a -dimensional vector subspace of sequences of independent random variables, such that for every nonzero we have for every , , and converges almost surely.
Proof. Let be a probability space such that there exists a sequence of independent random variables such that Clearly, and for any , so Let be an almost disjoint (i.e., an intersection of any two different members is finite) family of infinite subsets of . For and , define a random variable by the formula: and observe that and . Hence, for any and . Let be the subspace of spanned by . We will show that has the desired properties. Let be of the form , for some , , and . Clearly, is a sequence of independent random variables (by the independence of ) and for all . Moreover, for infinitely many naturals, namely, from the set , we have so In particular, is -dimensional vector space. As is almost disjoint, there is such that the sets are pairwise disjoint. Let and observe that therefore By the Borel-Cantelli Lemma so In particular, converges almost surely.
5. Failure of Fubini’s Theorem
The famous Theorem of Fubini states that for any integrable function the following equality holds: (from now on integrable, measurable, almost every (a.e.) means Lebesgue integrable, Lebesgue measurable, from a co-null set). The idea of looking for examples of functions (measurable unbounded or nonmeasurable) attracted an attention of many mathematicians (see, for example, [10–13]). In this section, we will present some lineability results (both positive and negative) in this area.
5.1. The Case of Bounded Functions
Let be a bounded function. We say that for the Iterated Integral Property Failure holds when
(IPPF)(i) is integrable for almost every , is integrable for almost every ,(ii), defined a.e., is integrable, , defined a.e., is integrable,(iii).
It is known that existence of a bounded function with (IIPF) is independent from (we refer the reader to  for the details). In this situation, we may obtain some results consistently with .
Theorem 14. It is consistent with that the set of bounded functions with (IIPF) is -lineable (or star-like) but not -lineable.
Proof. Let be the so-called Steinhaus set, i.e., a set with all horizontal sections of measure zero and all vertical sections of full measure (existence of such a set is consistent with ; for example, it can be constructed under the Continuum Hypothesis; see ). It is easy to see that, for the characteristic function of and any , the function has (IIPF).
To prove the second assertion, let be bounded and satisfy (IIPF). Denote (the real numbers) , and , in analogous way. Then . It is easy to see that for nonzero numbers , , and the function , we have so does not have (IIPF) and the proof is finished.
Remark 15. A modification of a function with (IIPF) on the set of measure zero still has (IIPF). Hence, there are distinct bounded functions with (IIPF).
5.2. The Case of Unbounded Functions
It is natural to ask a question about the (IIPF) condition for an unbounded function