Let $p$ be an odd prime, let $\mathbb{Z}_{2,p}$ denote $\mathbb{Z}_{2}\times\mathbb{Z}_{p}$, let $dx$ and $dy$ be the Haar probability measures on $\mathbb{Z}_{2}$ and $\mathbb{Z}_{p}$, respectively, and consider a function $F:\mathbb{Z}_{2,p}\rightarrow\mathbb{R}$ in $L^{1}\left(\mathbb{Z}_{2,p}\right)$.
The Fubini-Tonelli Theorem for abstract measure spaces tells us that the integrability of $F$ is sufficient to guarantee the existence of: $$G\left(y\right)=\int_{\mathbb{Z}_{2}}F\left(x,y\right)dx$$ for almost every $p$-adic integer $y$. Moreover, $G$ is in $L^{1}\left(\mathbb{Z}_{p}\right)$.
Using the translation-invariance of the Haar measure, I can prove that:
$$\lim_{N\rightarrow\infty}\int_{\mathbb{Z}_{2}}\frac{1}{2^{N}}\sum_{n=0}^{2^{N}-1}F\left(2x+n,y\right)dx=\int_{\mathbb{Z}_{2}}\left(\int_{\mathbb{Z}_{2}}F\left(2x+z,y\right)dz\right)dx=\int_{\mathbb{Z}_{2}}\left(\int_{\mathbb{Z}_{2}}F\left(z,y\right)dz\right)dx=G\left(y\right)$$
holds for almost every $p$-adic integer $y$ in the case where $F$ is continuous. Does this result hold (say, by the Dominated Convergence Theorem) when $F$ is merely integrable?