Articles of measure theory

Lebesgue-$\sigma$-algebras $\mathfrak L^{p+q}\neq\mathfrak L^p \otimes\mathfrak L^q$

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras $\mathfrak L$. So first of all, given a zero Lebesgue-null-set $N$ in $\mathbb R^p$, I have proven that $N\times B$ is a Lebesgue-null-set in $\mathbb R^{p+q}$ for arbitrary […]

Strictly monotone probability measure

Let $m$ be a probability measure on $X \subseteq \mathbb{R}^n$. Let $f: X \rightarrow \mathbb{R}$ be measurable. Assume that there exists $\epsilon > 0$ such $m\left( \{ x \in X \mid |f'(x)| \leq \epsilon \}\right) = 0$, that is, $f$ is almost flat on a set of measure $0$. Under what conditions, for all $\alpha\geq […]

Uncountable disjoint union of measure spaces

Let $(a,b)$ be an interval. Let $(A_i, \Sigma_i, \mu_i)$ be a measure space for each $i \in (a,b)$. Is it possible to put a measure space on the disjoint union $$\bigcup_{i \in (a,b)}\{i\}\times A_i?$$ If the $A_i = \Omega_i$ were open subsets of $\mathbb{R}^n$, we can think of this disjoint union as a non cylindrical […]

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given a Borel regular measure $ \mu $ in $\mathbb{R}^n $, given a $\mu$-measurable subset $E \subset \mathbb{R}^n $, let $$ \psi(x,E) = \lim_{r\rightarrow 0}\frac{\mu(E\cap B(x,r))}{\mu(B(x,r))} $$ Here […]

Existence of an extending measure

Let $\Omega$ be a nonempty set and $\cal{A}$ be any class of subsets of $\Omega$ including emptyset. Suppose that $\mu:\cal{A}\to R^{+}\cup{+\infty}$ be such a non-zero function that the equality $\mu(\cup_{n \in N}A_n)=\sum_{n \in N}\mu(A_n)$ holds true when $A_n \in \cal{A}$ for each $n \in N$, $\cup_{n \in N}A_n \in \cal{A}$ and $A_k \cap A_m=\emptyset$ for […]

Product measure with a Dirac delta marginal

Let $(S,\mathcal F)$ be a measurable space, and let $\nu \in\mathcal P(S,\mathcal F)$ be a probability measure on $(S,\mathcal F)$. Fix some $x\in S$ and consider Dirac measure $\delta_x$. Would like to prove If $\mu \in \mathcal P(S×S,\mathcal F\otimes \mathcal F)$ and has marginals $ν$ and $δ_x$ $then$ $μ=ν×δ_x$ So we should show $μ(A×B)=ν(A)δ_x(B)$ for […]

Show that this function is not continuous except on a set of measure zero

Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals Let $$g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$$ where $$\chi_{(0,1]} = \left\{\begin{array}{ll} 1&\mbox{if $x-r_n \in (0,1]$,}\\ 0&\mbox{otherwise.} \end{array}\right.$$ Show that $g$ is not continuous except possibly on a set of measure $0$.

Understanding answer to question showing equality of two measures on the Borel $\sigma$ algebra on $\mathbb{R}$

I understand that this question has been asked before, however my question is in reference to a specific answer given here. Proving two measures of Borel sigma-algebra are equal In particular it is the last answer (which the OP provided). Conveniently their answer has numbers so I when I am referring to numbers I am […]

If $\mathcal A$ generates $\mathcal S$ then $\sigma (X )=\sigma (X ^{-1 } ( \mathcal A ))$

Show that if $\mathcal A$ generates $\mathcal S$ then $X ^{-1 } ( \mathcal A )= \{\{X \in A \} : A \in \mathcal A \} $ generates $\sigma (X) =\{\{X \in B \} : B \in \mathcal S \}$ First I interpret this as $\sigma (X )=\sigma (X ^{-1 } ( \mathcal A ))$ […]

Infinite Lebesgue integral over all infinite measure subsets?

This question is in particular for $\mathbb{R}^+$. What properties must a finite function $f$ have such that $\int_A f d\mu = \infty$ for all A with $\mu(A) = \infty$? As pointed out previously (when the question was not framed correctly), obviously any constant $f$ will have infinite integral.