New posts in measure-theory

An outer measure is countable-additive on the measurable sets

real-analysis measure-theory lebesgue-measure

$f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable

Convergence in $L^p$ and convergence almost everywhere

functional-analysis measure-theory lp-spaces

Ergodic action of a group

measure-theory lie-groups ergodic-theory group-actions

Meaning of $\int_E {f(x) \mu(dx)}?$

measure-theory definition lebesgue-integral

Is D a borel subset?

real-analysis measure-theory

Example of Converge in measure, but not converge point-wise a.e.?

real-analysis measure-theory convergence-divergence examples-counterexamples

Inverse images and $\sigma$-algebras

How to show density of 2^a 3^b

Prove that $m^(A\cup B)=m^(A)+m^*(B)$ whenever $\exists \alpha>0$ such that $|a-b|>\alpha$ for any $a\in A,b\in B$

analysis measure-theory lebesgue-measure outer-measure

Dirac measures are extreme points of unit ball of $C(K)^*$.

real-analysis functional-analysis measure-theory

$f_n^\alpha(x) = n^\alpha x^n$ converges almost everywhere

measure-theory lebesgue-measure pointwise-convergence

About the Wasserstein "metric"

measure-theory probability-theory metric-spaces

About absolute continuity $\Rightarrow$ null set maps to null set

measure-theory lebesgue-integral

$L^2$ norm and $L^{\infty}$ inequality for periodic smooth functions

analysis measure-theory lebesgue-integral lp-spaces integral-inequality

The Multiplication Operator $M_f: L^2(\mu) \to L^2(\mu)$ such that $M_f g = fg$ (Rudin)

real-analysis functional-analysis measure-theory

Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$

real-analysis integration measure-theory convergence-divergence solution-verification

Proof question: Sequences of measurable functions $f_n$, such that for almost all $x$, set $f_n(x)$ is bounded...

measure-theory solution-verification

A function that is bounded and measurable but not Lebesgue integrable

real-analysis integration measure-theory lebesgue-measure examples-counterexamples

If $\ \sum_{k=1}^n m(E_n) > n-1,$ then prove that $\bigcap_{k=1}^n E_k$ has positive measure.

real-analysis measure-theory lebesgue-measure