Functional distribution monads in functional-analytic contexts

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 $\textit{functional-analytic context}$,formulatedintermsofagivenmonadoralgebraictheory$$\mathcal{T}$$enrichedinaclosedcategory$$\mathcal{V}$$. By employing the notion of $\mathit{\text{commutant}}$ for enriched algebraic theories and monads, we define the $\mathit{\text{functional}}\mathit{\text{distribution}}\mathit{\text{monad}}$ 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$-convex spaces}$foragivenpreorderedring$R$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.