New posts in lebesgue-measure

Completion of borel sigma algebra with respect to Lebesgue measure

measure-theory lebesgue-measure

Every subset of a subspace of $\mathbb{R}^n$ of dim $<n$ has measure 0

measure-theory lebesgue-measure

Showing that there do not exist uncountably many independent, non-constant random variables on $ ([0,1],\mathcal{B},\lambda) $.

probability-theory measure-theory random-variables lebesgue-measure independence

Does the equality $\lambda (\pi ^{-1}(B+g))=\lambda (\pi^{-1}(B))$ holds for any $B\in \mathfrak{B}_G$ and $g\in G$?

measure-theory lebesgue-measure borel-measures

How can I find a subset of a set with "half the size" of the original?

measure-theory lebesgue-measure

Outer measure of a nested sequence of non-measurable sets

real-analysis measure-theory lebesgue-measure

$f_n \rightarrow f$ & $|f_n|\le g\in L_1$ Prove: $f\in L_1$ | $\lim_{n\rightarrow\infty} \int_X f_n d\mu=\int_X f d\mu$ | $f_n\rightarrow f$ in $L_1$

real-analysis measure-theory lebesgue-integral lebesgue-measure

If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

general-topology measure-theory lebesgue-measure

Solution verification: $m(A)>1$ implies $\exists x,y\in A$ s.t. $x-y\in\mathbb{N}.$

real-analysis integration measure-theory solution-verification lebesgue-measure

Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?

real-analysis integration measure-theory lebesgue-integral lebesgue-measure

Every non-measurable $X \subseteq \mathbb{R}^n$ has non-measurable $Y \subseteq X$ such that $|Z|=0$ for every measurable $Z \subseteq Y$

measure-theory lebesgue-measure

Proving that a trivial product sigma algebra is the product of sigma algebras

real-analysis measure-theory lebesgue-measure measurable-sets

Prove $X(\omega) = \sup\{y \in \mathbb{R}: F(y) < \omega\}$ is a random variable.

probability-theory measure-theory probability-distributions lebesgue-measure quantile

Lebesgue measure of a subspace of lower dimension is 0

lebesgue-measure

Is there a set $A \subset [0,1]$ such that $\int_{A \times A^\text{c}} \frac{\mathrm{d} x \, \mathrm{d} y}{\lvert x - y\vert}=\infty$?

integration lebesgue-integral lebesgue-measure

For $f :\mathbb{R} \rightarrow \mathbb{R}\displaystyle$ we have ${\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere

real-analysis lebesgue-measure

Prove that $f\in L^1(A)\Leftrightarrow \sum_{n}^{\infty}m(\{ x\in A : f(x)\geq n \}) < \infty$

lebesgue-integral lebesgue-measure measurable-functions measurable-sets

Are all measure zero sets measurable?

real-analysis measure-theory lebesgue-measure

$\lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x$

limits functions lebesgue-integral lebesgue-measure pointwise-convergence

Lebesgue integral of a positive function on a set of positive measure

measure-theory lebesgue-integral lebesgue-measure