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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 $q\geq p$.

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    For b) do you have to show $\{F_p\mid p\in \mathbb{N}\}$ is closed ? Because if seems that $F_p$ is just an element of $M$. – 2017-02-10
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    Should be showed that $F_p$ is closed not $\{F_p| p \in \mathbb{N}\}$ – 2017-02-10
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    Oh, I see. $F_p$ is the intersections of families and hence a family of sets of $M$ itself. Can you elaborate on what you have tried so far? – 2017-02-10
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    I thought about show that $F_p$ contain all limit points. But, i don't know how begin the sketch. – 2017-02-10
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    Could it be that the $F_{ij}$ are closed already? Hence $F_p$ would be closed as an intersection of closed sets. – 2017-02-10
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    I will try this. Perhaps in this way work; perhaps using the fac that $\{v_k\}$ is Cauchy for each $E \in M$. – 2017-02-10

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