For $E \subset \mathbb{R}$ and $\epsilon >0$, $\exists$ $(a,b)$ s.t. $\theta(E \cap (a,b)) \geq (1-\epsilon)|b-a|$ ($\theta$ Lesbegue Outer Measure)
63413
For a set of positive measure there is an interval in which its density is high, $\mu(E\cap I)> \rho \mu(I)$
14048
Proof that the complement of zero-measure set on $[0,1]$ is dense in $[0,1]$
85818
$E(|X-Y|)$ for $X$, $Y$ i.i.d and uniform on $[0,1]$
51134
If $E \in \sigma(\mathcal{C})$ then there exists a countable subset $\mathcal{C}_0 \subseteq \mathcal{C}$ with $E \in \sigma(\mathcal{C}_0)$
60032
If the measure of union = sum of outer measures, then the sets are measurable
94971
For every $\epsilon>0$ there exists $\delta>0$ such that $\int_A|f(x)|\mu(dx) < \epsilon$ whenever $\mu(A) < \delta$
158498
$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
37206
$\mathbb{R}^n\times\{0\}$ has measure zero in $\mathbb{R}^{n+1}$
60110
The set of differences for a set of positive Lebesgue measure
136795
A non-measurable set of given outer measure
6450
Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing
108995
Prove $|e^{i\theta} -1| \leq |\theta|$
55019
Lebesgue density theorem in the line
78256
Finite additivity in outer measure
53239
Expectation inequality for a set and its subset.
72506
An example s.t. $E(E(X|\mathcal F_1)|\mathcal F_2)\neq E(E(X|\mathcal F_2)|\mathcal F_1)$
64320
Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$