### Archive

Posts Tagged ‘measurable’

## 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 ${\displaystyle \int_\Omega f_ng_n}$ where ${f_n}$ converges pointwise to ${f}$ and ${g_n}$ converges to ${g}$ in some fashion, but not pointwise. For example, we could have that ${g_n}$ converges to ${g}$ in ${L^1}$. In this case we could write the integral ${\displaystyle \int_\Omega f_ng_n}$ as ${\displaystyle \int_\Omega f d\mu_n}$ where ${\mu_n}$ is the measure defined by ${\mu_n(A) = \displaystyle\int_A g_n}$. All measures considered in this post will be positive measures.

Certain hypotheses on the measures ${\mu_n,\mu}$ 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 ${\mu_n,\mu}$, named setwise convergence, which will allow us to prove the lower semicontinuity result. We say that ${\mu_n}$ converges setwise to ${\mu}$ if ${\mu_n(A) \rightarrow \mu(A)}$ for every measurable set ${A}$. 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 ${\mu_n}$ be a sequence of measures defined on ${\Omega}$ which converges setwise to a measure ${\mu}$ and ${(f_n)}$ a sequence of nonnegative measurable functions which converge pointwise (or almost everywhere in ${\Omega}$) to the function ${f}$. Then

$\displaystyle \int_\Omega f d\mu \leq \liminf_{n \rightarrow \infty} \int_\Omega f_n d\mu_n.$

## Generalized version of Lebesgue dominated convergence theorem

October 10, 2014 1 comment

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 ${(f_n)}$ be a sequence of measurable functions defined on a measurable set ${\Omega}$ with real values, which converges pointwise almost everywhere to ${f}$. Suppose that there exists a sequence of positive, integrable functions ${g_n}$ such that the two following conditions hold:

• ${|f_n(x)|\leq g_n(x)}$ for almost every ${x \in \Omega}$;
• There exists an integrable function ${g}$ such that ${(g_n) \rightarrow g}$ in ${L^1(\Omega)}$.

Then

$\displaystyle \lim_{n \rightarrow \infty} \int_\Omega f_n = \int_\Omega f$

## Infinitely Countable Sigma Algebra

A famous result in measure theory is the following

There is no  infinitely countable $\sigma$-algebra.

This plainly states that if $S$ is a $\sigma$-algebra on a space $X$, then $S$ is finite or $card(S) \geq card(\Bbb{R})$.

## sets distance construction and properties

Denote by $\Sigma$ the quotient space of the family of Lebesgue measurable sets of $\Bbb{R}^N$ by the equivalence relation $E_1 \sim E_2 \Leftrightarrow \chi_{E_1}=\chi_{E_2} a.e.$. Denote by $|X|$ the Lebesgue measure of the measurable set $X$.
1) Prove that $\delta(E_1,E_2)=\arctan( |E_1 \Delta E_2|)$ is a distance on $\Sigma$.
2) Prove that given $(E_n)_{n \geq 1}, E$ measurable sets in $\Bbb{R}^N$ the following three properties are equivalent.

• $\delta(E_n,E) \to 0$;
• $\chi_{E_n}-\chi_E \xrightarrow{\sigma(L^1,L^\infty)} 0$;
• $\chi_{E_n}-\chi_E \xrightarrow{L^1} 0$.

3) Prove that $(\Sigma,\delta)$ is a complete metric space.

4) Given the sequence $(f_n)$ of integrable real valued functions on $\Bbb{R}^N$, such that for any measurable set $E$ of $\Bbb{R}^N$ there exists $\displaystyle \lim_{n \to \infty}\int_E f_n$, prove that if $|E| \to 0$ then $\displaystyle \sup_n\int_E |f_n| \to 0$.

Categories: Analysis, Real Analysis Tags:

## Positive measure set

February 2, 2011 1 comment

Prove that in any Lebesgue measurable set $A \subset \Bbb{R}^2$ such that $m(A)>0$, 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 $A$ 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 $A$. In short terms, $A$ contains a figure similar to any figure formed of finitely many points.

Categories: Uncategorized Tags:

## Measure zero

Suppose $f$ is integrable on $\mathbb{R}^d$. For each $\alpha >0$, let $E_\alpha=\{x : |f(x)| > \alpha\}$. Prove that $\int_{\mathbb{R}^d} |f(x)| dx = \int_0^\infty m(E_\alpha) d\alpha$.