Let $(X, \mathcal{M}, \mu)$ be a measure space, $\mu (X) = 1$ and $f \colon X \rightarrow \mathbb{R}$ measurable such that $f \in L^2(\mu)$, $\int_{X}f d\mu =0$ and $\int_{X}f^2 d\mu = 1$. Show that $\mu(\{x \in X \colon f(x)> \alpha\}) \leq 1/(1 + \alpha^2)$ for all $\alpha >0$
I've been trying to solve this problem for many hours, but nothing good has come of it. I restrict myself to uploading my attempts because they do not add up to anything interesting, instead I would like to know a simple and less cumbersome test like my ideas and improve my reasoning. I appreciate any response you can give me.