How will Chebychev inequality look like incase of a p-norm.

Question

If I have this as a given:

$$\lim _{n \rightarrow \infty }\int_{E} |f_{n} - f|^p = 0.$$

And I want to use Chebychev inequality to show convergence in measure of $\{f_{n}\}.$ How will Chebychev inequality look like? and why?

Answer 1

\begin{align*} \epsilon^{p}|(|f_{n}-f|\geq\epsilon)|\leq\int|f_{n}-f|^{p}\rightarrow 0, \end{align*} so \begin{align*} |(|f_{n}-f|\geq\epsilon)|\rightarrow 0. \end{align*}