  We give a general categorical construction that yields several monads of
measures and distributions as special cases, alongside several monads of
filters. The construction takes place within a categorical setting for
generalized functional analysis, called a $\textit{functional-analytic
context}$, formulated in terms of a given monad or algebraic theory
$\mathcal{T}$ enriched in a closed category $\mathcal{V}$. By employing the
notion of $\textit{commutant}$ for enriched algebraic theories and monads, we
define the $\textit{functional distribution monad}$ associated to a given
functional-analytic context. We establish certain general classes of examples
of functional-analytic contexts in cartesian closed categories $\mathcal{V}$,
wherein $\mathcal{T}$ is the theory of $R$-modules or $R$-affine spaces for a
given ring or rig $R$ in $\mathcal{V}$, or the theory of $\textit{$R$-convex
spaces}$ for a given preordered ring $R$ in $\mathcal{V}$. We prove theorems
characterizing the functional distribution monads in these contexts, and on
this basis we establish several specific examples of functional distribution
monads.
