A class of harmonic function spaces is introduced and studied, namely the spaces $H^p_\sigma(G)$ of $\sigma$-harmonic $L^p$ functions on a locally compact group $G$, for $1\leq p \leq \infty$ and a given complex measure $\sigma$ on $G$ of unit norm. It is shown that there is a contractive projection from $L^p(G)$ onto $H^p_\sigma(G)$, for $1\lt p\leq \infty$, and structural results for $H^p_\sigma (G)$ are deduced. Given an adapted probability measure $\sigma$ on $G$, a uniqueness result is proved, that the space $H^p_\sigma(G)$ contains only constant functions, for $1\leq p\,{\lt}\,\infty$. For any $\sigma$, a result on the dimension of $H^1_\sigma (G)$ is proved.