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Let $G$ be a discrete group. Consider $G^{\mathbb{N}}$, the space of all sequences in $G$ equipped with product $\sigma$-algebra. Let $m:G^{\mathbb{N}} \to G^{\mathbb{N}}$ be the multiplication map defined by $$m(w)_n=w_1w_2\ldots w_n$$ The push-forward measure $\mathbb{P}_{\mu}=m_{*}\mu^{\mathbb{N}}$, where $\mu$ is a probability measure on $G$, is called the Markovian measure. In particular, $$\mathbb{P}_{\mu}\left(E=\{(w_i): w_1=g_1,w_2=w_1g_2,\ldots,w_n=w_{n-1}g_n\}\right)=\mu(g_1)\mu(g_2)\ldots\mu(g_n) $$ When equipped with $\mathbb{P}_{\mu}$, we think of $G^{\mathbb{N}}$ as the space of walks, where $w_n$ is the position of the random walk at the time $n$. We denote the space of walks by $(\Omega,\mathbb{P}_{\mu})$. A function $h: G \to \mathbb{R}$ is called $\mu$-harmonic if $$h(g)=\sum_{\gamma}\mu(\gamma)h(g\gamma)$$ We denote the space of all bounded harmonic functions by $H^{\infty}(G,\mu)$. Let $h$ be a bounded harmonic function. I want to show the following:

$(i)$ $\lim_nh(w_n)$ exists for $\mathbb{P}_{\mu}$-a.e $w \in \Omega$.

$(ii)$ Hence, for any $h \in H^{\infty}(G,\mu)$ corresponds to some $\tilde{h} \in L^{\infty}(\Omega,\mathbb{P}_{\mu})$ and $$h(e)=\int_{\Omega}\tilde{h}(w)d\mathbb{P}_{\mu}(\omega)$$

$(iii)$ Moreover, for a fixed $g$ the limit $\lim_{n \to \infty}h(gw_n)=\tilde{h}(g,w)$ exists and $$h(g)=\int_{\Omega}\tilde{h}(g,w)d\mathbb{P}_{\mu}$$

For $(i)$: Let $\mathcal{B}_n$ be the $\sigma$-algebra generated by the cylinder sets $C_{g_1,g_2,\ldots,g_n}=\{w \in \Omega: w_i=g_i \text{ for } i=1,2,\ldots,n\}$, i.e. $$\sigma_n=\langle\{C_{g_1,g_2,\ldots,g_n}: g_1,g_2,\ldots,g_n \in \Gamma\} \rangle $$

Then the $\sigma$-algebra on $\Omega$ is generated by $\cup_n\mathcal{B}_n$. We wish to show that the sequence of functions $f_n$ defined by $f_n(w)=h(w_n),f_0=h(e)$ forms a bounded martingale w.r.t$\{\mathcal{B}_n\}$. Towards that end, one checks that $$\mathbb{E}(f_{n+1}|\mathcal{B}_n)=\sum_{g \in G}\mu(g)h(w_ng)=h(w_n)=f_n(w)$$ The second equality comes from the fact that $h$ is a bounded harmonic function and the third equality comes from definition of $f_n$. I have trouble understanding how the first equality comes, I know that it comes from the way conditional expectation is defined but I am not seeing it. From the above equation, we see $\{f_n\}$ forms a bounded Martingale and hence, by Martingale Convergence Theorem, it converges a.e. I am not sure how $(ii)$ should be proven. I think $(iii)$ follows from $(ii)$, but not sure why.

Any hints for these would be greatly appreciated. Thanks for the help!!

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