Let $(X, \mathcal{A}, \mu)$ be a measurable space. I want to prove that is $f: X \longrightarrow \mathbb{R}$ is measurable, non-negative and its integral on $X$ is finite, then $\forall a>0$ the set $$A = \{x ∈ X : f(x) ≥ a\}$$ is $\mu-$finite (that is, $\mu (A) < \infty$). I think I could use this lemma to prove this statement but I don't really know how to do it. Could someone help me?
Asked
Active
Viewed 96 times
0
-
5Does this answer your question? Prove that $f\in L^1(A)\Leftrightarrow \sum_{n}^{\infty}m({ x\in A : f(x)\geq n }) < \infty$ – Alejandro Bergasa Alonso Jan 14 '21 at 20:43
