Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure says that the linear functional
$$\Phi: C_c(G) \rightarrow C_c(G/H)$$
$$\Phi(f)(gH) = \int\limits_H f(gh)dh \tag{1}$$
is surjective and its kernel contains the kernel of the functional $f \mapsto \int\limits_G f(g)dg $ on $C_c(G)$. Therefore, there is a unique linear functional $\Psi$ on $C_c(G/H)$ such that $\Psi \circ \Phi(f) = \int\limits_G f(g)dg$. By the Riesz representation theorem, $\Psi$ must be given by a measure on $G/H$, and this measure satisfies
$$\int\limits_G f(g)dg = \int\limits_{G/H} \int\limits_H f(gh)dh dg \tag{2}$$
for all $f \in C_c(G)$.
My question is, what can be said about (2) holding for more general functions, not necessarily compactly supported? Say, $f$ continuous and in $L^1(G)$. It isn't obvious that (1) even converges in this case.
If $f$ is continuous and integrable, and (1) converges for all $g \in G$, does (2) hold?