How to proof that convergence on metric hold iff convergence on measure
2239
Showing that $\{f_n \}$ converges to $f$ is equivalent to $\lim_{n\to \infty} \int_X \frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu(x)=0$
86230
$f_n \rightarrow f$ in measure and $g$ be measurable and finite a.e. Prove that $f_ng \rightarrow fg$ in measure
18429
Counting measure proof
95831
Compactness and sequential compactness in metric spaces
123972
Proving that every function defined on a discrete metric space is uniformly continuous.
156295
Relax Egoroff's Theorem to pointwise convergence a.e. and bounded a.e. pointwise limit
52104
A pseudometric on the space of the measurable functions is complete
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
9355
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
16890
Does convergence in probability imply a.s. convergence in a countable space?
94438
Show that the countable product of metric spaces is metrizable
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
70106
diameter on a compact metric space
60282
Prove that $f_n$ converges uniformly on $\mathbb{R}$
26369
Proving seemingly obvious equations with measure theory
71682
Proving that $X$ is a Banach space iff convergence of $\sum\|x_n\|$ implies convergence of $\sum x_n$