I have updated the LaTeX to WordPress Converter to change \[ \] to $l atex\displaystyle and $ respectively.

Note that \[ … \] is preferable over $$ … $$.

Test code:

Input:

If $(X,\Sigma,\mu)$ is a measure space, $f$ is a non-negative measurable extended real-valued function, and $\epsilon>0$, then \[\mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu.\]

Define \[s(x)=\begin{cases}

\epsilon, &\text{if}\ f(x)\geq\epsilon\\

0, &\text{if}\ f(x)<\epsilon.

\end{cases}\]

Then $0\leq s(x)\leq f(x)$. Thus $\int_X f(x)\,d\mu\geq\int_X s(x)\,d\mu=\epsilon\mu(\{x\in X: f(x)\geq\epsilon\})$. Dividing both sides by $\epsilon>0$ gives the result.

Output:

If is a measure space, is a non-negative measurable extended real-valued function, and , then

Define

Then . Thus . Dividing both sides by gives the result.

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