Uniform convergence in measures

**ster** · December 9th 2009, 06:05 AM

Just stuck on a Measure theory question right now:

Suppose $\{f_n\}\subset {L^1} and {f_n}\longrightarrow f$ uniformly. Show that if $\mu(X)<\infty , then f\in {L^1}$ and $\int f_n\longrightarrow \int f$ . Find an example that shows that the result is not necessarily true when $\mu(X)= \infty$

Little bit stumped on how to start. Initially I thought I could use the dominated convergence theorem, but not fully sure. We do have uniform convergence and the dominated convergence theorem claims if we can bound a function by g then the $\int f =lim_{n\rightarrow\infty} \int {f_n}$ .

If not I'm thinking of using chebychev's inequality.

Any help would be great.

I also apologise for any problems with the LaTex, it's my first time using it.

Thanks

**Jose27** · December 11th 2009, 09:15 AM

Since $f_n \rightarrow f$ uniformly we have that for all $x\in X$ there exists an $N$ such that $m,n\geq N$ implies $\vert f_n(x)\vert - \vert f_m(x) \vert \leq \vert f_m(x)-f_n(x) \vert \leq 1$ now taking $m=N$ we get $\vert f_n(x) \vert \leq 1+ \vert f_N(x)\vert$ for all $n\geq N$ and this last one is integrable since $\int_{X} d\mu =\mu (X)<\infty$ so picking $g(x)=\max \{ \vert f_1(x) \vert ,...,\vert f_{N-1}(x) \vert , 1+ \vert f_N(x) \vert \}$ we get our dominating function.

When $\mu (X)= \infty$ see here

**GTO** · December 21st 2009, 10:34 PM

looks like what u used cauchy sequence rather than uniformly convergence. can you still do what you did with uniform convergent sequence?

**putnam120** · December 21st 2009, 11:07 PM

$|\int_Xf_n-f|\le\int_X|f_n-f|\le\mu(X)\cdot\sup|f_n-f|<\epsilon\mu(X)$ .

For a counter example consider $f_n=\frac{1}{n}\chi_{[0,n]}$ . Then $f_n\to 0$ but $\int_\mathbb{R}f_n=1$ for all $n$ .

**PRLM** · December 21st 2009, 11:37 PM

can you show me how to show that $f$ is in L1, please?

**putnam120** · December 22nd 2009, 09:10 AM

I think that the following version of the triangle inequality should help: $|a|-|b|\le |a-b|$

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