Folland question 15(a) Prove that $\mu(E):=\sup\{\mu(F)\mid F\subset E,\ \mu(F)<\infty\}$ is semifinite
139540
Real Analysis, Folland Problem 1.3.15 Measures
149820
How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed?
32965
Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$
147705
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
142812
Prove that $f\in L^1(A)\Leftrightarrow \sum_{n}^{\infty}m(\{ x\in A : f(x)\geq n \}) < \infty$
130916
Proving the identity functions is continuous iff $\tau_1\supseteq\tau_2$
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
116966
Prove $\limsup_{n \rightarrow \infty} a_n\geq1$ if $\lim_{n \rightarrow \infty} a_na_{n+1}=1$
157426
Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
93684
Show that $\mu (E \setminus A)>0$ if $A$ is a nonmeasurable subset of $X$ and $A \subseteq E$, $E$ is measurable.
78763
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
150896
Show that $\lim_{p \to \infty} \| f \|_p = \| f \|_\infty$
43168
Prove that $\limsup\left(b_{n}a_{n}\right)=\limsup\left(a_{n}\right) $ when $\lim_{n\rightarrow\infty}\left(b_{n}\right)=1$
149362
Prove that $ \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = \int_{- \infty}^\infty \int_{- \infty}^\infty (F(x,y)-F_X(x) F_Y(y)) \,dx\,dy,$
92588
Showing that $(C_0([0,\infty)), \|\cdot\|_\infty)$ is complete.