Definition:Marginal of Measure on Product Space

Definition

Let $\struct {X_1, \Sigma_1}$ and $\struct {X_2, \Sigma_2}$ be measurable spaces.

Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra.

Let $\pi$ a measure on $\Sigma_1 \otimes \Sigma_2$.

The marginal of $\pi$ on $X_i$ for $i \in \set {1,2}$ is the measure $\mu_i$ on $\Sigma_i$ defined as:

$\forall A \in \Sigma_i : \map {\mu_i} A := \map \pi {\set { \struct {x_1, x_2} \in X_1 \times X_2 : x_i \in A } }$

Sources

• 2003: Cédric Villani: Topics in Optimal Transportation: $1.$ Formulation of the optimal transportation problem