Articles of measure theory

Convergence of Lebesgue integrals

I am sitting on this multiple-choice question and I cannot answer it, nor say if it is right or wrong: Given non-negative, Lebesgue-integrable functions $f,f_k\colon E\rightarrow \mathbb{R}^+$ with $\displaystyle\forall x \in E\setminus N: \lim_{k \to \infty}f_k(x)=f(x)$, where $\lambda(N)=0$, $E,N \subset \mathbb{R}^n$ and $\displaystyle\lim_{k \rightarrow \infty}\int_E f_k(x) d\lambda=\int_E f(x) d\lambda$. Is it always true that $$\lim_{k […]

The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$

I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. Let me try to elucidate my understanding of the topic, in the hope that somebody patient and kind might be able to fill in the […]

measurable functions and existence decreasing function

Let $f(t)$ be a measurable and almost everywhere finite function, defined on the closed interval $E = [a, b]$. Prove the existence of a decreasing function $g (t)$, defined on [a, b], which satisfies the relation $m(E \cap \left \{ x: {g > x} \right \}) = m(E \cap \left \{ x: {f > x} […]

Comparing the Lebesgue measure of an open set and its closure

Let $E$ be an open set in $[0,1]^n$ and $m$ be the Lebesgue measure. Is it possible that $m(E)\neq m(\bar{E})$, where $\bar{E}$ stands for the closure of $E$?

Proving the measure of an increasing sequence of measurable sets is the limit of the measures

Show that if $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ is an increasing sequence of measurable sets (so $A_j\subseteq A_{j+1}$ for every positive integer $j$), then we have $$m\left(\bigcup_{j=1}^\infty A_j\right)=\lim_{j\to\infty}m(A_j)$$ Here is my proof: According to the $\sigma$-algebra property, $\bigcup_{j=1}^{\infty}A_j$ is a measurable set, so it makes sense to talk about $m(\bigcup_{j=1}^{\infty}A_j)$. Firstly I prove that $\lim_{j\to\infty}m(A_j)\leq m(\bigcup_{j=1}^{\infty}A_j)$. This […]

Does the everywhere differentiability of $f$ imply it is absolutely continuous on a compact interval?

Suppose $f$ is differentiable everywhere on $[0,1]$. Must $f$ be absolutely continuous on $[0,1]$? I know this is true if $f’$ is integrable but I’m not sure in this more general case.

If $E \in \sigma(\mathcal{C})$ then there exists a countable subset $\mathcal{C}_0 \subseteq \mathcal{C}$ with $E \in \sigma(\mathcal{C}_0)$

Given a collection of sets $\mathcal{C}$ and $E$ an element in the $\sigma$-algebra generated by $\mathcal{C}$, how do I show that $\exists$ a countable subcollection $\mathcal{C_0} \subset \mathcal{C}$ such that $E$ is an element of the $\sigma$-algebra, $\mathcal{A}$ generated by $\mathcal{C_0}$? The hint says to let $H$ be the union of all $\sigma$-algebras generated by […]

Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$

I have the following question about $L^p$ spaces: Suppose that $f,f_1,f_2,\ldots$ are functions in $L^p$ for some $p \geq 1$ and the sequence converges in $L^p$ to $f$, i.e. $||f_n-f||_p \to 0$. Does this imply that the sequence $|f_1|^p,|f_2|^p,\ldots$ converges in $L^1$ to $|f|^p$? Is it also true if we remove the absolute value? That […]

Is integration by substitution a special case of Radon–Nikodym theorem?

I was wondering if Integration by substitution is a method only for Riemann integral? if Integration by substitution is a special case of Radon–Nikodym theorem, and why? Thanks and regards!

Fatou's lemma and measurable sets

I don’t know how can I imply Fatou’s lemma for any measurable sets $A_k$ that is.. $\lambda(\liminf A_k)\le\liminf\lambda(A_k)$ how can I prove it? and is there any example in $R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? thx for your help!.