can we make Birkhoff's Ergodic Theorem uniform?

Question

Let $(X , \mu)$ be a probability space and $T : X \to X$ be a measure preserving transformation such that $\mu$ is ergodic with respect to $T$. We know, that for every $\varphi \in L^1 (X , \mu)$ the time average converges to the space average almost everywhere, i.e.$$\frac{1}{n}\sum_{k=0}^{n-1} \varphi\big(T^k (x) \big) \xrightarrow{n \to \infty} \int_X \varphi ~ d \mu \quad \text{for a.e. } x\in X.$$Can we make this uniform in the following sense? Let $X' \subset X$ be the set of all $x \in X$ s.th. the above limit exists. Now given $\varepsilon >0$, can we find a set $X_1\subset X'$ with $\mu(X_1) > 1 - \varepsilon$ and such that there is $N \in \mathbb N$ so that $$\left\lvert \frac{1}{n}\sum_{k=0}^{n-1} \varphi\big(T^k (x) \big) - \int_X \varphi ~ d \mu \right\rvert < \varepsilon$$ holds for all $x \in X_1$ and all $n \ge N$?