We define a random measure generated by a real anisotropic harmonizable fractional stable field $Z^H$ with stability parameter $α\in(1,2)$ and Hurst index $H\in(1/2,1)$ and prove that the measure is $σ$-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in $\mathbb R^3$ with a random source generated by $Z^H$. We show that the solution to this equation, given by Kirchhoff's formula, has a modification, which is Hölder continuous of any order up to $(3H-1)\wedge 1$. In the case where $H\in(2/3,1)$, we show further that the modification is absolutely continuous.