Articles of measure theory

continuity and measure

Let $ A, B \subseteq \mathbb {R} $ be Lebesgue measurable sets such that at least one of them has finite measure. Let $ f $ be the function defined by $$f (x) = m ((x + A) \cap B)$$ for each $ x \in \mathbb{R} $. Show that $ f $ is continuous. Hint: […]

natural problem where a Lebesgue integral turns out to be useful

I’d like to know if there are examples of problems that don’t “outwardly” seem to require the powerful tools of measure theory, but whose solutions nonetheless require (or, say, are very greatly simplified by) proper use of measure theory, and in particular Lebesgue integration. An analogy might be the use of complex numbers in mathematics; […]

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to \infty} f_{n\, j}(t), \qquad \text{for all }t\in I $$ and $f_{n\, j}\in C(I)$. Question. Is it true that there exists a sequence $g_n\in C(I)$ such that […]

Question from Folland Chapter 1 Exercise 14

The problem is the following: If $\mu$ is a semifinite measure and $\mu(E)=\infty$, for any $C>0$ there exists $F \subset E$ with $C < \mu(F) < \infty$. We were told as a hint to consider the set $F = \left\lbrace F\subset E : 0 < \mu(F)<\infty \right\rbrace$. I know that because $\mu$ is semifinite, we […]

Poincaré Inequality – Product Of Measures

I’m given two euclidean spaces $ \mathbb{R}_1 , \mathbb{R}_2 $ , with probability measures on them , that satisfy the Poincar√©’s inequality: $ \lambda^2 \int_{\mathbb{R}^k} |f – \int_{\mathbb{R}^k} f d\mu | ^2 d\mu \leq \int_{\mathbb{R}^k} | \nabla f | ^2 d\mu $ for some constants $C_1 , C_2 $ respectively. How can I prove that […]

Image of $A\subset \mathbb R^d$ under a Lipschitz function is $H^d$ measurable.

I really need an help with the following exercise. Suppose that $A\subseteq \mathbb R^d$ is Lebesgue measurable. Let $f\colon A \to \mathbb R^k $ be a Lipschitz function. Show that $f(A)$ is $H^d$ measurable in $\mathbb R^k$, where $H^d$ is the $d-$ dimensional Hausdorff measure. Now the main problem is that I don’t know how […]

Show that the product $\sigma $-algebra is generated by the $\pi $-system $\{B _1 \times B _2 : B \in \Sigma _1 , B _2 \in \Sigma _2 \} $

Show that the product $\sigma $-algebra is generated by the $\pi $-system $\{B _1 \times B _2 : B_1 \in \Sigma _1 , B _2 \in \Sigma _2 \} $ Let $(X , \Sigma _1) $ and $(Y , \Sigma _2 ) $ be two measure spaces Let $\Sigma = \Sigma _1 \times \Sigma _2$ […]

Equivalence of definition for weak convergence

The Wikipedia page on convergence of measures says: In the case $S=\mathbb{R}$ with its usual topology, if $F_n, F$ denote the cumulative distribution functions of the measures $P_n$, $P$ respectively, then $P_n$ converges weakly to $P$ if and only if $\lim_{n\rightarrow\infty} F_n(x) = F(x)$ for all points $x\in\mathbb{R}$ at which $F$ is continuous. I’m interested […]

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I’d like to prove that for all $a_1,a_2\in(0,1)$ and $t\in[0,1]$ the following holds: \begin{align} F(t a_1+(1-t)a_2)\leq t F(a_1)+(1-t)F(a_2)\end{align} using the properties of the logarithm yields $F(a)=a \log\left(\int_X \lvert f\lvert^{1/a}\right)$. Plugging this […]

Bochner Integral: Approximability

Problem Given a measure space $\Omega$ and a Banach space $E$. Consider a Bochner measurable function $S_n\to F$. Then it admits an approximation from nearly below: $$\|S_n(\omega)\|\leq \vartheta\|F(\omega)\|:\quad S_n\to F\quad(\vartheta>1)$$ (This is sufficient for most cases regarding proofs.) Can it happen that it does not admit an approximation from below: $$\|S_n(\omega)\|\leq\|F(\omega)\|:\quad S_n\to F$$ (I’m just […]