Product measure

Remarks

Existence follows from Carathéodory's extension theorem applied to the rectangles, and uniqueness from σ-finiteness and the π-λ theorem. For $E \in \Sigma \otimes T$ one has $(\mu \times \nu)(E) = \int_X \nu(E_x) \, d\mu(x) = \int_Y \mu(E^y) \, d\nu(y)$, where $E_x$ and $E^y$ are the sections of $E$.

Example: let $\mu$ and $\nu$ be Lebesgue measure on the Borel sets of $[0, 1]$. Then $\mu \times \nu$ is two-dimensional Lebesgue measure on the Borel sets of $[0, 1]^2$: both measures give each rectangle its area, so they agree by uniqueness.

Non-example: let $\delta$ be the image of Lebesgue measure $\lambda$ on $[0, 1]$ under $x \mapsto (x, x)$, a measure on the diagonal of $[0, 1]^2$. For $A = [0, \tfrac12]$ and $B = [\tfrac12, 1]$, $\delta(A \times B) = \lambda(A \cap B) = 0$, but $\lambda(A) \lambda(B) = \tfrac14$. So $\delta$ is not $\lambda \times \lambda$.

Without σ-finiteness, uniqueness can fail. For Lebesgue measure on $[0, 1]$ and counting measure on $[0, 1]$, there are different measures on the product σ-algebra with the rectangle property. The product of Lebesgue measures is often completed to give Lebesgue measure on $\R^{m+n}$.