Variations on Fatou’s Lemma – Part 2

As we have seen in a previous post, Fatou’s lemma is a result of measure theory, which is strong for the simplicity of its hypotheses. There are cases in which we would like to study the convergence or lower semicontinuity for integrals of the type where converges pointwise to and converges to in some fashion, but not pointwise. For example, we could have that converges to in . In this case we could write the integral as where is the measure defined by . All measures considered in this post will be positive measures.

Certain hypotheses on the measures allow us to find a result similar to Fatou’s Lemma for varying measures. In the following, we define a type of convergence for the measures , named setwise convergence, which will allow us to prove the lower semicontinuity result. We say that converges setwise to if for every measurable set . The following proof is taken from Royden, H.L., Real Analysis, Chapter 11, Section 4. It is very similar to the proof of Fatou’s lemma given here.

Theorem A. Let be a sequence of measures defined on which converges setwise to a measure and a sequence of nonnegative measurable functions which converge pointwise (or almost everywhere in ) to the function . Then

Generalized version of Lebesgue dominated convergence theorem

The following variant of the Lebesgue dominated convergence theorem may be useful in the case we can not dominate a sequence of functions by only one integrable function, but by a convergent sequence of integrable functions.

Let be a sequence of measurable functions defined on a measurable set with real values, which converges pointwise almost everywhere to . Suppose that there exists a sequence of positive, integrable functions such that the two following conditions hold:

• for almost every ;
• There exists an integrable function such that in .

Then

Infinitely Countable Sigma Algebra

A famous result in measure theory is the following

There is no  infinitely countable -algebra.

This plainly states that if is a -algebra on a space , then is finite or .

sets distance construction and properties

Denote by the quotient space of the family of Lebesgue measurable sets of by the equivalence relation . Denote by the Lebesgue measure of the measurable set .
1) Prove that is a distance on .
2) Prove that given measurable sets in the following three properties are equivalent.

• ;
• ;
• .

3) Prove that is a complete metric space.

4) Given the sequence of integrable real valued functions on , such that for any measurable set of there exists , prove that if then .

Positive measure set

Prove that in any Lebesgue measurable set such that , there exists three points in the set such that they form a triangle similar with a given one.

It is possible to generalize this in the following way: Consider any configuration of finitely many points. If has positive measure, then there exists a dilation and a translation (and eventually a rotation) such that these transformations applied to the given configuration makes it a part of . In short terms, contains a figure similar to any figure formed of finitely many points.

Measure zero

Prove that any subset of the real line having Lebesgue measure equal to zero can be translated into the irrationals.

Integral formula

Suppose is integrable on . For each , let . Prove that .

Measurable finite-valued function

Let be a measurable finite-valued function on , and suppose that is integrable on . Show that is integrable on .

Measurable sets

Let be a sequence of measurable subsets of the real line which covers almost every every point infinitely often. Prove that there exists a set of zero density () such that also covers every point infinitely often.
Miklos Schweitzer 2009 Problem 8