So my problem is:
Let $f \in L^1$. Prove lim$_{n\rightarrow \infty} n*m(\{a:|f(a)|\geq n\})=0$.
How would I go about proving this?
So my problem is:
Let $f \in L^1$. Prove lim$_{n\rightarrow \infty} n*m(\{a:|f(a)|\geq n\})=0$.
How would I go about proving this?
Consider the function $\phi_n(x) = n \; 1_{\{t \; | \; |f(t) | \geq n \}}(x)$, and note that $|\phi_n(x)|\leq |f(x)|$ for all $n$ and $x$. Furthermore, for almost all $x$, we have $\lim_{n \rightarrow \infty} \phi_n(x) = 0$. Consequently we can use the dominated convergence theorem to conclude that $\lim_{n \rightarrow \infty} \int\phi_n = 0$. Since $\int\phi_n = n \; \mathbb{m} \{t \; | \; |f(t) | \geq n \}$, we have the desired result.