Let \((\Omega,\Sigma; P)\) be a probability space, \(T :\Omega\to\Omega\) a measure-preserving transformation, \(X\) an integrable random variable, and \(\operatorname{Im}\) the \(\sigma\)-field of invariant sets. Denote \[ S_k :=\sum_{i=0}^{k-1} X\circ T^i, \;\;M := \sup_{k\in N} S_k. \] In the present paper, the author first proves the Maximal Ergodic Inequality \[ \int_{[M>0]}X dP\geq 0 \] in a short proof simpler than that of M. Keane and K. Petersen [Easy and nearly simultaneous proofs of the ergodic theorem and maximal ergodic theorem, Dynamics & Stochastics, Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 248–251 (2006; Zbl 1121.37301)]. Then this inequality is generalized as follows: \[ \int_{A\cap[\sup_{k\in N}\frac{S_k}{k}>Y]} X dP\geq \int_{A\cap [\sup_{k\in N}\frac{S_k}{k}>Y]} Y dP \] for any invertible integrable random variable \(Y\) and \(A\in\operatorname{Im}\). Finally, an application of this inequality yields the Pointwise Ergodic Theorem.