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The product axiom for the Giry monad is given as follows:

$$ \mu_{X}: P(P(X)) \to P(X) $$

given by

$$ \mu_X (M)(A) := \int_{P(X)} \tau(A) M(d\tau). \,. $$

$P(X)$ is equipped with the weakest topology which makes the integration map $\tau \mapsto \int_{X}f d\tau$ continuous function for any $f$, a bounded, continuous, real function on $X$.

The list monad is easy to understand because we have simple explanations like "a list of lists goes to a list by concatenation". Can someone give a nice explanation of the product for the Giry Monad in the same spirit as that for the List monad?

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    $\begingroup$ A probability measure on an affine space has an average value (also called expectation or integral), which is a point in that affine space. Apply this to the affine space of probability measures on $X$. In other words, a measure on the space of measures determines a (weighted) average of those measures. $\endgroup$ – Andreas Blass Sep 1 '18 at 22:26

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