the following is causing me trouble:
Let $(Ω_i, F_i, \mu_i),(i = 1, 2)$ be two finite measure spaces. Then for any $A \in F1 ⊗ F2, x_1 → \mu_2(A_{x_1})$ (resp. $x_2 → \mu_1(A^{x_2})$) is measurable on $(Ω_1, F_1) (resp. (Ω_2, F_2))$ and we have $$\int_{Ω_1}\mu_2(A_{x_1})\mu_1(dx_1)=\int_{Ω_1}\mu_1(A^{x_2})\mu_2(dx_2)$$
This is a lemma from the lecture notes and we've previously defined $$A_{x_1} = \{x_2 ∈ Ω_2 : (x_1, x_2) \in A\}$$ and $$A^{x_2} = \{x_1 ∈ Ω_1 : (x_1, x_2) \in A\}$$
The problem is I can't really wrap my head around the $\mu_1(dx_1)$ and $\mu_2(dx_2)$ bits. It's different from the $d\mu_1$ and $d\mu_2$ notation we've used so if someone can help me I'd be grateful.