A sequence converging uniformly to 0 so that Fatou's Lemma is a strict inequality

Question

I have to find a sequence of measurable functions such that it converges uniformly to zero but Fatou's Lemma is a strict inequality.

I can find a sequence of functions that converges pointwise to $0$ so that Fatou's Lemma is strict. However, when I want it converges uniformly to zero, I got stuck.

Please help me. Any advice is highly appreciated.

Answer 1

For $f_n=\frac1n\chi_{(0,n)}$ we have $f_n\rightarrow0$ uniformly since $$\left|f_n-f_m\right|<\frac1n+\frac1m<\frac2N$$for $n,m>N$.

Then $$0=\int_{\mathbb{R}}\liminf_{n\rightarrow\infty}f_{n}dm<\liminf_{n\rightarrow\infty}\int_{\mathbb{R}}f_{n}dm=1$$and we get strict inequality in Fatou's lemma.