  A previous study analyzed the convergence of probability densities for
forward and inverse problems when a sequence of approximate maps between model
inputs and outputs converges in $L^\infty$. This work generalizes the analysis
to cases where the approximate maps converge in $L^p$ for any $1\leq p <
\infty$. Specifically, under the assumption that the approximate maps converge
in $L^p$, the convergence of probability density functions solving either
forward or inverse problems is proven in $L^q$ where the value of $1\leq
q<\infty$ may even be greater than $p$ in certain cases. This greatly expands
the applicability of the previous results to commonly used methods for
approximating models (such as polynomial chaos expansions) that only guarantee
$L^p$ convergence for some $1\leq p<\infty$. Several numerical examples are
also included along with numerical diagnostics of solutions and verification of
assumptions made in the analysis.
