Riesz-Markov type-theorem in ergodic theorem

Question

I was reading some ergodic theory-related material and I came across the following result.

Theorem: Let $C_{b}(\mathbb{R})$ the space of real-valued bounded continuous functions on $\mathbb{R}$, equipped with the supremum norm $\|f\|_{\infty} := \sup_{x\in \mathbb{R}}|f(x)|$. There exists a positive linear functional $\lambda: C_{b}(\mathbb{R}) \to \mathbb{R}$, such that: $$\lambda(\varphi) = \langle \varphi \rangle := \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^{T}\varphi(t)dt \tag{1}\label{1}$$ for every $\varphi \in C_{b}(\mathbb{R})$ where this limits exists. Moreover, $\lambda$ satisfies the following properties:

1. $\lambda(\varphi) \ge 0$ for all $\varphi \in C_{b}(\mathbb{R})$ in which the limit (\ref{1}) exists.
2. $\lambda(1) = 1$
3. $\lambda(\varphi_{s}) = \lambda(\varphi)$, where $\varphi_{s}(t) := \varphi(t-s)$, $t,s \in \mathbb{R}$.

I apologize if this question is too basic, but I have absolutely no background in ergodic theory and I have no idea if this result is standard in the theory or not. In any case, I would like to know how to prove it. It seems one needs some sort of Riesz-Markov argument to prove it, but the translation invariance (property 3.) and formula (\ref{1}) are beyond me. Any help in finding a proof of this theorem is welcome. Thanks!

Answer 1

Thanks to the hints given in the comments, I now think I can answer my own question.

First, set: $$\mathcal{M} = \{\varphi \in C_{b}(\mathbb{R}): \mbox{$\langle \varphi\rangle := \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^{T}\varphi(t)dt$ exists}\},$$ which is easily seen to be a vector subspace of $C_{b}(\mathbb{R})$. Notice that $\varphi \equiv 1 \in \mathcal{M}$ and: $$\langle 1 \rangle = 1. \tag{1}\label{1}$$ Now, define $\lambda: \mathcal{M}\to \mathbb{R}$ by $\lambda(\varphi) := \langle \varphi\rangle$. This is a positive linear functional on $\mathcal{M}$, which satisfies: $$|\lambda(\varphi)| \le \|\varphi\|_{\infty}, \tag{2}\label{2}$$ from where it follows that $\|\lambda\| \le 1$. By (\ref{1}) and the definition of the norm $\|\lambda\|$, we obtain $\|\lambda\| = 1$. Hence, by the Hahn-Banach Theorem, one can extend $\lambda$ to a positive linear functional on $C_{b}(\mathbb{R})$, with the same norm. The translation invariance (property 3. of the post) follows from the argument given in the comments. Alternatively, it follows from the change of variables formula for the Riemann integral.