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

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

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

I wish to prove that for any classifer $ g:X\rightarrow\{0,1\}$ it holds that
$ 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,  $L_D(h)=D\{(x,y)| h(x)\not = y\}$.

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