Alternatively a version of this theorem (and a proof) can be found in Einsiedler & Ward's Ergodic Theory: with a view towards Number Theory as Theorem 4.10 on p. 105 (let $\mathcal{E}^T(X)$ be the set of extremal points of the set $\mathcal{M}^T(X)$ of $T$-invariant Borel probability measures on $X$, which we know by Kryloff-Bogoliouboff to be non-empty):

Theorem: If $X$ is a compact metric space and $T\in C^0(X,X)$, then TFAE:

1. $T$ is uniquely ergodic.
2. $\vert \mathcal{E}^T(X)\vert=1$ (as ergodic measures are precisely the extremal points of invariant measures).
3. $\forall f\in C^0(X,\Bbb{R}),\exists C_f\in\Bbb{R},\forall x\in X:\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^n(x)\to C_f$.
4. $\forall f\in C^0(X,\Bbb{R}),\exists C_f\in\Bbb{R}:\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^n\stackrel{u.}{\to} C_f$.
5. 3. Holds in a dense set of $C^0(X,\Bbb{R})$.

Alternatively a version of this theorem (and a proof) can be found in Einsiedler & Ward's Ergodic Theory: with a view towards Number Theory as Theorem 4.10 on p. 105:

Let $\mathcal{E}^T(X)$ be the set of extremal points of the set $\mathcal{M}^T(X)$ of $T$-invariant Borel probability measures on $X$, which we know by Kryloff-Bogoliouboff to be non-empty. As ergodic measures are precisely the extremal points of invariant measures, $\mathcal{E}$ stands both for extremal and ergodic.

Theorem: If $X$ is a compact metric space and $T\in C^0(X,X)$, then TFAE:

1. $T$ is uniquely ergodic.
2. $\vert \mathcal{E}^T(X)\vert=1$.
3. $\forall f\in C^0(X,\Bbb{R}),\exists C_f\in\Bbb{R},\forall x\in X:\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^n(x)\to C_f$.
4. $\forall f\in C^0(X,\Bbb{R}),\exists C_f\in\Bbb{R}:\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^n\stackrel{u.}{\to} C_f$.
5. 3. Holds in a dense set of $C^0(X,\Bbb{R})$.