In mathematics, the Giry monad is a construction that assigns to a measurable space a space of probability measures over it, equipped with a canonical sigma-algebra. It is one of the main examples of a probability monad.

It is implicitly used in probability theory whenever one considers probability measures which depend measurably on a parameter (giving rise to Markov kernels), or when one has probability measures over probability measures (such as in de Finetti's theorem).

Like many iterable constructions, it has the category-theoretic structure of a monad, on the category of measurable spaces.

The Giry monad, like every monad, consists of three structures:

Let ${\displaystyle (X,{\mathcal {F}})}$ be a measurable space. Denote by ${\displaystyle PX}$ the set of probability measures over ${\displaystyle (X,{\mathcal {F}})}$. We equip the set ${\displaystyle PX}$ with a sigma-algebra as follows. First of all, for every measurable set ${\displaystyle A\in {\mathcal {F}}}$, define the map ${\displaystyle \varepsilon _{A}:PX\to \mathbb {R} }$ by ${\displaystyle p\longmapsto p(A)}$. We then define the sigma algebra ${\displaystyle {\mathcal {PF}}}$ on ${\displaystyle PX}$ to be the smallest sigma-algebra which makes the maps ${\displaystyle \varepsilon _{A}}$ measurable, for all ${\displaystyle A\in {\mathcal {F}}}$ (where ${\displaystyle \mathbb {R} }$ is assumed equipped with the Borel sigma-algebra).

Equivalently, ${\displaystyle {\mathcal {PF}}}$ can be defined as the smallest sigma-algebra on ${\displaystyle PX}$ which makes the maps

measurable for all bounded measurable ${\displaystyle f:X\to \mathbb {R} }$.

The assignment ${\displaystyle (X,{\mathcal {F}})\mapsto (PX,{\mathcal {PF}})}$ is part of an endofunctor on the category of measurable spaces, usually denoted again by ${\displaystyle P}$. Its action on morphisms, i.e. on measurable maps, is via the pushforward of measures. Namely, given a measurable map ${\displaystyle f:(X,{\mathcal {F}})\to (Y,{\mathcal {G}})}$, one assigns to ${\displaystyle f}$ the map ${\displaystyle f_{*}:(PX,{\mathcal {PF}})\to (PY,{\mathcal {PG}})}$ defined by