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Posts Tagged ‘measure’

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

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

Characterization of sigma-finite measure spaces

Let ${(X,\mathcal{E})}$ be a measure space and ${\mu}$ be a positive measure on it; prove that ${L^1(X,\mu)}$ contains a strictly positive function if and only if ${X}$ is ${\sigma}$-finite with respect to ${\mu}$.

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

Categories: Measure Theory, Real Analysis Tags:

The Brunn-Minkowski Inequality

May 9, 2012 1 comment

The Brunn-Minkowski Inequality gives an estimate for the measure of the set ${A+B=\{a+b : a \in A,b \in B\}}$ in terms of the Lebesgue measures of ${A}$ and ${B}$ in ${\Bbb{R}^d}$ (of course, we can only speak of such an inequality if all the sets ${A,B,A+B}$ are Lebesgue measurable). The inequality has the form

$\displaystyle |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}.$

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 $card(S) \geq card(\Bbb{R})$.

Measurable set has nice property.

Let $S \subset [0,1]^2$ be a measurable set such that $m(S)>1/2$ (in the Lebesgue measure). Prove that there exist $x,y,z$ such that $(x,y),(x,z),(y,z) \in S$.

MathOverflow

Reverse Riesz Type Problem

Suppose $p \in (1,\infty)$ and consider $q$ such that $1/p+1/q=1$. Take $(X,\mathcal{A},\mu)$ a $\sigma$-finite measure space. Suppose $f:X \to \Bbb{C}$ is a measurable function such that $fg \in L^1(\mu)$ for all $g \in L^q(\mu)$.

Prove that $f \in L^p(\mu)$.

Categories: Measure Theory Tags:

p-integrable product

December 13, 2010 1 comment

Let $(X,\mathcal{M},\mu)$ be a fixed measure space, and $p_i,\ 1\leq i\leq n$ be positive nmbers such that $p_i >1,\ 1\leq i\leq n$ and $\displaystyle \sum_{i=1}^n\frac{1}{p_i}=\frac{1}{p}$, with $p>1$ (works for $p=1$ also).
For each $i \in \{1,2,...,n\}$ consider $f_i \in \mathcal{L}^{p_i}(\mu)$. Is it true that $f_1f_2...f_n \in \mathcal{L}^p(\mu)$?
PHD 4302