In a paper that I am reading, the author uses measures as initial data for pdes. The setting is as following: Let $\Omega\subset \mathbb{R}^n$ be a bounded open set with smooth boundary. Let $u_0$ be a nonnegative measure on $\Omega$. Let $u$ be the solution of the problem \begin{align} u_t - \Delta u &= f(x,t) & &\text{on } \Omega\times(0,\infty),\\ u&=0 & &\text{on }\partial\Omega\times(0,\infty), \\ u &= u_0 & &\text{for a.e. $x$ in $\Omega$}, \end{align} where $f$ is a globally Lipschitz continuous function.
The author uses the properties of $u$, as I would if $u_0\in L^\infty(\Omega)$. For instance he even uses the convolution of $u_0$ with the Green function for the heat equation, i.e., $$\int_\Omega G(x,y,t) u_0(y) dy.$$
But I don't even know how to think of a solution $u$ where $u_0$ is a measure. How do I show that such a solution $u$ exists, is regular, etc. Any help would be appreciated.