Example of an integral not converging

Question

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable and integrable and $\int f_n\, d\mu\to\int f\, d\mu$. Find an example that shows that the condition $\mu(\Omega)<\infty$ is necessary.

Hello, I already proved the claim, but I did not find a counterexample yet that shows, that $\mu(\Omega)<\infty$ cannot be neglected.

Can you help me to find one?

My first idea was to consider $(\mathbb{R},\mathcal{B},\lambda)$, where $\lambda$ is the Lebesgue-measure. Anyway, it is $\lambda(\mathbb{R})=\infty$.

Answer 1

Take on the measure space you suggest $f_n(x):=\frac 1n\chi_{(0,n)}$.

Answer 2

On the real line, how about this: $$f_n(x)=\frac{\chi_{[-n,n]}(x)}{1+|x|}$$