Question: Let $(X,M, \mu)$ a finite measure space and define a metric space $M'$ as follows: for $A,B \in M$, define: $d(A,B) = \mu(A\triangle B)$.
The space $M'$ is defined as all sets in $M$ where sets $A$ and $B$ are identified if $\mu(A\triangle B) = 0$.
Let $\{v_k\}$ be a sequence of measures on the finite measure space $(X, M, \mu)$ such that
$\bullet v_k(X) < \infty$ for each $k$,
$\bullet $the limit exists and is finite for each $E\in M$ $v(E) := \displaystyle\lim_{k \to\infty} v_k(E)$,
$\bullet v_k << \mu$ for each $k$.
a)Prove that each $v_k$ is continuous on the metric space $(M', d)$.
b)For each $\epsilon > 0$ let $F_{i,j}:=\{E \in M; |v_i(E) - v_j(E)| \leq \displaystyle\frac{\epsilon}{3}\}$ $i, j = 1, 2, ... $ and
$F_p:= \displaystyle\bigcap_{i, j \geq p} F_{i, j}$ , $p = 1, 2,...$
Prove that $F_p$ is a closed set in $(M', d)$.
The item a) is obtained from the theorem of Radon-Nikodym: For each $k$ there is a $f_k \in L^1(\mu)$ such that $v_k(E) = \displaystyle\int_{E} f_k d\mu$, hence $v_k$ is continuous. This is right?
But for th item (b) i have not any idea. Any tip?
Note that $F_p \subset F_q, q\geq p$; moreover if $A \subset M$ is limit of sequence $\{E_j\}$, $E_j \in F_p$, then $A \in F_q$ for some $q\geq p$.