Let $(X, \mathcal{A}, \mu)$ be a measurable space and f a non negative measurable function. Let $E=\{x \in E: f(x) < 1\}$. Calculate $$\lim_{n \to \infty} \int_E e^{-f^n} d\mu.$$
I'm having some trouble in exercises on integrability. I know that because $f < 1$, $$\lim_{n \to \infty} e^{-f^n} = e^0 = 1.$$ What can I do else?
Consider the sequence $g_n = e^{-f^n}$. Note that $(g_n\chi_E)_{n\geq 1}$ is an increasing sequence of non-negative measurable functions, so you can use the Monotone convergence theorem. The limit of the integrals is $\mu(E)$.