= Solution
We first prove invariance, before using any assertion that requires it. Write $A_n\varphi(x)=n^{-1}\sum_{k=0}^{n-1}\varphi(f^kx)$. Choose a point whose forward <orbit> has the prescribed convergence against every <continuous function>; the hypothesis gives a set of such points of full <measure>. For each <continuous function> $\varphi$, telescoping gives
$$
A_n(\varphi\circ f)(x)-A_n\varphi(x)
=\frac{\varphi(f^nx)-\varphi(x)}n\longrightarrow0,
$$
because a <continuous function> on a <compact metric space> is bounded. Both test functions are continuous, so their averages converge to their prescribed integrals. Hence
$$
\int_X\varphi\circ f\,d\mu=\int_X\varphi\,d\mu
$$
for every <continuous function> $\varphi$. The continuous-test criterion for an <invariant measure> therefore gives \b[$f_*\mu=\mu$]. The points in the hypothesis can now be called <generic points for an invariant measure>.
For <ergodicity>, let $B$ be a <Borel set> invariant modulo null sets, and set $h=\mathbf1_B$, $p=\mu(B)$. Then $h\circ f=h$ almost everywhere. Since invariance preserves null preimages, deleting the union of all iterated preimages of the exceptional <null set> gives $h(f^kx)=h(x)$ for every $k\geq0$ on one full-measure set. Thus $A_nh=h$ almost everywhere for every $n$.
Given $\epsilon>0$, the <continuous approximation of Borel indicators on a compact metric space>, proved in the supporting steps below, supplies $\varphi\in C(X)$ with $0\leq\varphi\leq1$ and $\|h-\varphi\|_{L^1(\mu)}<\epsilon$. By the hypothesis, $A_n\varphi\to\int\varphi\,d\mu$ almost everywhere. Since $0\leq A_n\varphi\leq1$, the <dominated convergence theorem> also gives convergence in $L^1(\mu)$. Invariance and the <triangle inequality> give the uniform estimate
$$
\|A_n(h-\varphi)\|_{L^1(\mu)}
\leq\frac1n\sum_{k=0}^{n-1}\int_X|h-\varphi|\circ f^k\,d\mu
=\|h-\varphi\|_{L^1(\mu)}<\epsilon.
$$
Consequently,
$$
\begin{aligned}
\|h-p\|_{L^1(\mu)}
&\leq\|A_nh-A_n\varphi\|_{L^1(\mu)}
+\left\|A_n\varphi-\int\varphi\,d\mu\right\|_{L^1(\mu)}
+\left|\int\varphi\,d\mu-p\right|\\
&<2\epsilon+\left\|A_n\varphi-\int\varphi\,d\mu\right\|_{L^1(\mu)}.
\end{aligned}
$$
Letting $n\to\infty$ and then $\epsilon\downarrow0$ proves $\|h-p\|_1=0$. Explicitly,
$$
\|\mathbf1_B-p\|_1=p(1-p)+(1-p)p=2p(1-p),
$$
so $p\in\{0,1\}$. Every invariant <Borel set> has <measure> zero or one, which proves
$$
\boxed{\mu\text{ is invariant and ergodic}.}
$$
This proof controls measurable-set averages through a uniform $L^1$ estimate; convergence only against continuous test functions is not silently assumed to include <indicator functions>.
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