Let $(X,\mathcal{A})$ and $(X,\mathcal{B})$ be measurable spaces. Every measure $\mu$ on $(X\times Y, \mathcal{A}\otimes\mathcal{B})$ gives an assignment $\mathcal{A}\times\mathcal{B}\to\Bbb{R}$ via $(A,B)\mapsto\mu(A\times B)$.
Conversely, consider an assignment $\mathcal{A}\times\mathcal{B}\to\Bbb{R}$. Under which conditions does this extend to a measure on $(X\times Y, \mathcal{A}\otimes\mathcal{B})$?