Answer by egreg for How to put equations inside brackets and define their labels?

I guess you want empheq. However:

1. \Big should be either \Bigl or \Bigr;
2. \exp doesn't take an argument;
3. \mathrm{Inf} should be an operator;
4. split was, in any case, useless.

\documentclass{article}\usepackage{amsmath,amssymb,empheq}\DeclareMathOperator{\prob}{\mathbb{P}}\DeclareMathOperator{\Inf}{\Inf}\begin{document}\begin{empheq}[left=\empheqlbrace]{align}& \prob_1 \Bigl[\Inf_{A,1}^{+}(h^*) > \frac{1}{2} + \frac{\epsilon}{2} \Bigr]  \leq \exp\Bigl(- \frac{d \epsilon^2}{18}\Bigr)  \label{res1plus1}\\[0.5ex]& \prob_1 \Bigl[\Inf_{A,0}^{-}(h^*) > \frac{1}{2} + \frac{\epsilon}{2} \Bigr]  \leq \exp\Bigl(- \frac{d \epsilon^2}{18}\Bigr)  \label{res1minus0}\\[0.5ex]& \prob_1 \Bigl[\Inf_{A,0}^{+}(h^*) > \frac{1}{2} - \frac{\epsilon}{2} \Bigr]  \leq \exp\Bigl(- \frac{d \epsilon^2}{2}\Bigr)  \label{res1plus0}\\[0.5ex]& \prob_1 \Bigl[\Inf_{A,1}^{-}(h^*) > \frac{1}{2} - \frac{\epsilon}{2} \Bigr]  \leq \exp\Bigl(- \frac{d \epsilon^2}{2}\Bigr)  \label{res1minus1}\end{empheq}\end{document}