Help in understanding this, convergence in measure.

Question

Let $(X,A,\mu)$ be a measurable space. $g,f_1,f_2\ldots \in L^1(\mu)$ , $g\geq 0$. $f:X\to C$ measurable. Suppose $f_n$ converges to $f$ in measure ($\mu$) and $\forall n\in N |f_n|\leq g$.

Prove that :

$lim_{n\to \infty} \int_{X} f_n d\mu = \int_X f d\mu$.

I'm trying to show this with using the dominated convergence theorem of Lebesgue since we have almost all the conditions . But we have to show that $f_n$ converges to $f$ almost everywhere. From the given information we can conclude that $f_n$ is a couchy sequence So by another theorem there is a sub sequence $f_{n_j}$ such that $f_{n_j}$ converges to $f$ almost everywhere. I don't know how to finish it.Though I know that there is a theory in calculus which states: $a_n$ converges iff every subsequent of $a_n$ has a subsequent that converges. But I don't know how to use it.

Can you help in this, and explain how it can be showed.

Answer 1

Prove it by contradiction. Suppose it is not true that $\int f_n d\mu \to \int f d\mu$. Then there exists $\epsilon >0$ and integers $1 \leq n_1<n_2<....$ such that $|\int f_{n_k} d\mu- \int f d\mu| \geq \epsilon$ for all $k$ $\,\,\, \cdots(1)$.

Now $(f_{n_k})$ converges to $f$ is measure because $f_n \to f$ in measure. Hence $(f_{n_k})$ has a subsequence $(f_{n_{k_i}})$ converging almost everywhere to $f$. By DCT we get $\int f_{n_{k_i}} d\mu \to \int fd\mu$ But this contradicts (1).

Answer 2

As you've mentioned, a sequence $(x_n)$ in a topological space X converges to a point $x\in X$ if and only if any subsequence of $(x_n)$ has a further subsequence that converges to $x$.

Now, let $(f_n)_{n \in M \subset \mathbb N}$ be a subsequence of $ (f_n)_{n \in \mathbb N}$. We wish to show that there exists a further subsequence $(f_n)_{n \in L\subset M}$ such that $$ \left (\int f_{n} \right)_{n \in L} \to \int f .$$ Evidently, $(f_n)_{n \in M} $ is dominated by $g$ and it converges to $f$ in measure. Thus, there exists a subsequence $(f_n)_{n \in L \subset M}$ such that $(f_n(x))_{ n \in L} \to f(x)$ for almost all x. By L.D.C theorem, we get the desired.