We define a class of spaces $H_{\mu }^p$, 0 < p < ∞, of holomorphic functions on the tube, with a norm of Hardy type: ${\left|\right|F\left|\right|}_{H_{\mu }^p}^p=su{p}_{y\in \Omega }{\int }_{\Omega ̅}{\int }_{ℝⁿ}{\left|F(x+i(y+t))\right|}^{p}dxd\mu \left(t\right)$. We allow μ to be any quasi-invariant measure with respect to a group acting simply transitively on the cone. We show the existence of boundary limits for functions in $H_{\mu }^p$, and when p ≥ 1, characterize the boundary values as the functions in $L_{\mu }^p$ satisfying the tangential CR equations. A careful description of the measures μ when their supports lie on the boundary of the cone is also provided....