Let $\mathcal{H}$ be a binary hypothesis class. The following are equivalent: 1. $\mathcal{H}$ is PAC learnable (uniform convergence holds for $\mathcal{H}$). 2. $\mathcal{H}$ has finite VC dimension. 3. The growth function $s(\mathcal{H}, n)$ is bounded polynomially in $n$. 4. The empirical risk minimizer $\hat{h}$ over $\mathcal{H}$ satisfies $\mathbb{E}[R(\hat{h})] - R(h^*) \to 0$ as $n \to \infty$ for any distribution $P_0$. Moreover, when $\operatorname{VC}(\mathcal{H}) = d < \infty$, the generalization bound takes the form \begin{align*} R(\hat{h}) \leq \min_{h \in \mathcal{H}} R(h) + C\sqrt{\frac{d \log n}{n}} \end{align*} with high probability, for a universal constant $C > 0$.