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We give a general categorical construction that yields several monads ofmeasures and distributions as special cases, alongside several monads offilters. The construction takes place within a categorical setting forgeneralized functional analysis, called a $\textit{functional-analyticcontext}$, formulated in terms of a given monad or algebraic theory$\mathcal{T}$ enriched in a closed category $\mathcal{V}$. By employing thenotion of $\textit{commutant}$ for enriched algebraic theories and monads, wedefine the $\textit{functional distribution monad}$ associated to a givenfunctional-analytic context. We establish certain general classes of examplesof functional-analytic contexts in cartesian closed categories $\mathcal{V}$,wherein $\mathcal{T}$ is the theory of $R$-modules or $R$-affine spaces for agiven ring or rig $R$ in $\mathcal{V}$, or the theory of $\textit{$R$-convexspaces}$ for a given preordered ring $R$ in $\mathcal{V}$. We prove theoremscharacterizing the functional distribution monads in these contexts, and onthis basis we establish several specific examples of functional distributionmonads.

Functional distribution monads in functional-analytic contexts