## 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

## Characterization of sigma-finite measure spaces

Let be a measure space and be a positive measure on it; prove that contains a strictly positive function if and only if is -finite with respect to .

*Ambrosio et al., Functions of Bounded Variation and Free Discontinuity Problems, Ex 1.5*

## The Brunn-Minkowski Inequality

The Brunn-Minkowski Inequality gives an estimate for the measure of the set in terms of the Lebesgue measures of and in (of course, we can only speak of such an inequality if all the sets are Lebesgue measurable). The inequality has the form

## 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 .

## Measurable set has nice property.

Let be a measurable set such that (in the Lebesgue measure). Prove that there exist such that .

*MathOverflow*

## Reverse Riesz Type Problem

Suppose and consider such that . Take a -finite measure space. Suppose is a measurable function such that for all .

Prove that .

## p-integrable product

Let be a fixed measure space, and be positive nmbers such that and , with (works for also).

For each consider . Is it true that ?

*PHD 4302*

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