Consider domain $X$, label set $ Y=\{0,1\}$ and the zero-one loss.

Given any probability distribution D over $ X\times \{0,1\} $, we've defined the Bayes classifier $ f_D $ to be-

$$ f_{D}(x)=
\begin{cases}
1 & \text{if }\mathbb{P}[y=1|x]\geq\tfrac{1}{2}\\
0 & \text{otherwise.}
\end{cases} $$

I wish to prove that, for any classifer $ g\colon X\rightarrow\{0,1\}$,
$ L_D(f_D)\leq L_D(g)$, which means that $ f_D$ is optimal.

$L_D(h) $ is defined to be the "true error" of the classifier $h$. That is,  $L_D(h)=D\{(x,y)\mid h(x)\not = y\}$.

I'm having some hard time proving this given the definitions above, and some hints/intuition will be appreciated.