Lebesgue Integrable functions

Question

I am in need of guidance for the following question: Let $f:X\to\mathbb{R}$ be an integrable function. Show that $\mu(\{x:|f(x)|\geq n\})\leq 1/n\int |f|\mu(dx)$ for each $n>0$.

Answer 1

Notice that for any $n$,

$$\int_X |f| d\mu \geq \int_{\{x: |f(x)|\geq n \}} |f| d\mu \geq \int_{\{x: |f(x)|\geq n \}} n \;d\mu = n \cdot \mu (\{x: |f(x)|\geq n \})$$

This is just Markov's inequality: http://en.wikipedia.org/wiki/Markov%27s_inequality#Statement