  We study the existence of solutions of the nonlinear problem $$ \left\{
\begin{alignedat}{2} -\Delta u + g(u) & = \mu & & \quad \text{in } \Omega,\\ u
& = 0 & & \quad \text{on } \partial \Omega, \end{alignedat} \right. $$ where
$\mu$ is a Radon measure and $g : \mathbb{R} \to \mathbb{R}$ is a nondecreasing
continuous function with $g(0) = 0$. This equation need not have a solution for
every measure $\mu$, and we say that $\mu$ is a good measure if the Dirichlet
problem above admits a solution. We show that for every $\mu$ there exists a
largest good measure $\mu^* \leq \mu$. This reduced measure has a number of
remarkable properties.

Analysis of PDEs;Functional Analysis