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Subset of plane with measure $1$ in all lines
measure-theory
set-theory
lebesgue-measure
Let $\alpha \in (0,1)$. Find a Borel subset $E$ of $[-1,1]$ s.t. $\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\alpha.$
real-analysis
measure-theory
lebesgue-measure
Interplay of Hausdorff metric and Lebesgue measure
measure-theory
lebesgue-measure
Let $f : \mathbb{R} \to \mathbb{R}$ be measurable and let $Z = {\{x : f'(x)=0}\}$. Prove that $λ(f(Z)) = 0$.
derivatives
lebesgue-measure
measurable-functions
Does the graph of a measurable function always have zero measure?
real-analysis
measure-theory
lebesgue-integral
lebesgue-measure
Set $E\subset \mathbb{R}^n$ of positive Lebesgue measure such that the Lebesgue measure of its boundary is zero
measure-theory
lebesgue-measure
Non measurable subset of a positive measure set
measure-theory
lebesgue-measure
$E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$
real-analysis
measure-theory
lebesgue-measure
Composition of measurable & continuous functions, is it measurable?
real-analysis
analysis
measure-theory
lebesgue-measure
Equivalent ideas of absolute continuity of measures
real-analysis
analysis
measure-theory
lebesgue-integral
lebesgue-measure
If $f'(x)>0$ on $E$ , where $m(E)>0,$ then $m(f(E))>0$
analysis
measure-theory
lebesgue-measure
$f$ a real, continuous function, is it measurable?
real-analysis
measure-theory
lebesgue-measure
measurable-functions
How to prove that if $f$ is continuous a.e., then it is measurable.
real-analysis
measure-theory
lebesgue-measure
simple-functions
Proving the *Caratheodory Criterion* for *Lebesgue Measurability*
real-analysis
measure-theory
lebesgue-measure
measurable-sets
Finite additivity in outer measure
measure-theory
lebesgue-measure
Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero
real-analysis
measure-theory
lebesgue-integral
lebesgue-measure
The Lebesgue measure of zero set of a polynomial function is zero
real-analysis
polynomials
lebesgue-measure
analyticity
analytic-functions
Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$
real-analysis
analysis
lebesgue-integral
lebesgue-measure
Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful
real-analysis
integration
measure-theory
lebesgue-integral
lebesgue-measure
Show that if the integral of function with compact support on straight line is zero, then $f$ is zero almost everywhere
real-analysis
integration
lebesgue-integral
lebesgue-measure
lp-spaces
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