Notation for Lebesgue Iterated Integral

Question

For double integrals, usually we will write $\int\dots\,dx\,dy$.

What if we are using Lebesgue measure $\lambda$ notation?

I have seen $\int\dots\lambda(dx)\lambda(dy)$, is this the correct notation? Are there other notations?

Answer 1

So if $(\Omega_1,\mathcal{F}_1,\lambda_1)$ and $(\Omega_2,\mathcal{F}_2,\lambda_2)$ are two measure spaces then we form the product space $(\Omega_1 \times \Omega_2)$ with product $\sigma$-algebra $\mathcal{F}_1 \times \mathcal{F}_2$ then there is a unique measure, say $\lambda_1 \times \lambda_2$ on $\mathcal{F}_1 \times \mathcal{F}_2$ such that $$ \lambda_1 \times \lambda_2 (A_1 \times A_2 ) = \lambda_1(A_1) \lambda_2(A_2) $$ for $A_1 \in \mathcal{F}_1$ and $A_2 \in \mathcal{F}_2$. And then the abstract result that lets you write $$ \int X d(\lambda_1 \times \lambda_2) = \int \left( \int X d\lambda_2 \right)d\lambda_1 = \int \int X d\lambda_2 d\lambda_1 $$ is Fubini's theorem.