Newbetuts

If $(X_n)_{n\in \mathbb{N}}$ is a martingale s.t. $\sup_n E[|X_n|]\leq M < \infty$, then $\sum_{n\geq 2}(X_n-X_{n-1})^2<\infty$ almost surely.

Limit a.e. of a sequence measurable functions is measurable

Can we really compose random variables and probability density functions?

$f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Function $f$ such that $|f(x)-f(y)| \ge \sqrt{|x-y|}$

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

A rigorous meaning of "induced measure"?

Is every compact subset of $\Bbb{R}$ the support of some Borel measure?

Is every $G_\delta$ set the set of continuity points of some function $f$? [duplicate]

cardinality of the Borel $\sigma$-algebra of a second countable space

Monotone convergence theorem by Fatou's lemma

Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$

$\exists$ countably generated $\mathcal F$, s.t. $\sigma(\{ \{\omega \}: \omega\in\Omega \}) \subsetneqq \mathcal F \subsetneqq \mathcal B(\Omega)$?

Is every vector space basis for $\mathbb{R}$ over the field $\mathbb{Q}$ a nonmeasurable set?

Are sets constructed using only ZF measurable using ZFC?

Measure of the irrational numbers?

New posts in measure-theory

Is the lattice of subspaces of a finite-dimensional scalar product space distributive?

Limit of norm $L^p$ when $p\to 0$ [duplicate]

Algebra generated by countable family of sets is countable?

Two sets $X,Y \subset [0,1]$ such that $X+Y=[0,2]$

If $(X_n)_{n\in \mathbb{N}}$ is a martingale s.t. $\sup_n E[|X_n|]\leq M < \infty$, then $\sum_{n\geq 2}(X_n-X_{n-1})^2<\infty$ almost surely.

Limit a.e. of a sequence measurable functions is measurable

Can we really compose random variables and probability density functions?

$f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Function $f$ such that $|f(x)-f(y)| \ge \sqrt{|x-y|}$

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

A rigorous meaning of "induced measure"?

Is every compact subset of $\Bbb{R}$ the support of some Borel measure?

Is every $G_\delta$ set the set of continuity points of some function $f$? [duplicate]

cardinality of the Borel $\sigma$-algebra of a second countable space

Monotone convergence theorem by Fatou's lemma

Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$

$\exists$ countably generated $\mathcal F$, s.t. $\sigma(\{ \{\omega \}: \omega\in\Omega \}) \subsetneqq \mathcal F \subsetneqq \mathcal B(\Omega)$?

Is every vector space basis for $\mathbb{R}$ over the field $\mathbb{Q}$ a nonmeasurable set?

Are sets constructed using only ZF measurable using ZFC?

Measure of the irrational numbers?