Strict inequality in Fatou lemma
= Strict inequality in Fatou lemma
On $([0,1],\lambda)$, the functions
$$
f_n=n\mathbf1_{(0,1/n)}
$$
converge pointwise to zero but satisfy $\int f_n\,d\lambda=1$. Hence the two sides of <Fatou lemma> can be zero and one.