Uniform Convergence Theorem: Let $f_k \in L(E)$ for $k=1,2,...$, and let ${f_k}$ converge uniformly to $f$ on $E$, $|E|<\infty$. Then $f\in L(E)$ and $\int_E f_k \rightarrow \int_E f $.
I need to show this fails if $|E|=\infty$. I have been trying to find a sequence of functions that converge uniformly over $\mathbb{R}$ but whatever I chose wasn't lebesgue integrable, I get the lebesgue integral is $\infty$.
Can someone give a hint? Thanks
Choose $f_n= \frac1n \chi_{(0,n)}$. Then, $f_n \rightarrow f$ uniformly, where $f \equiv 0$.
We have that for all $n$, $\int_{\Bbb R} f_n d \lambda = \frac1n \lambda((0,n)) = 1$, so $(f_n) \subset L^1(\Bbb R)$.
But $\int_{\Bbb R} f d \lambda = 0$ and $\lim \int_{\Bbb R} f_n d \lambda = 1$.