# Conditional expectation of a bounded harmonic function

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!!