Existence of a Set s.t. the Integral of a Non-negative Function Concentrates on It

Question

Let $(X,\mathcal{M},\mu)$ be a measure space. Show that if $f\in L^+$ (measurable and non-negative functions) and $\int fd\mu<\infty$ then, $\forall \varepsilon >0$ $\exists E\in\mathcal{M}$ s.t. $\mu(E)<\infty$ and $\int_E fd\mu>\int f d\mu-\varepsilon$

Answer 1

Let $E_k = \{x \in X : f(x) > k^{-1}\}$. Then $\mu(E_k) < \infty$ by Chebyshev's inequality, and $$\int f \, d\mu = \int_{\cup E_k} f \, d\mu\ = \lim_{k \to \infty} \int_{E_k} f \, d\mu$$ by e.g. the dominated convergence theorem.