Let $\mu$ be a probability measure on $\mathcal{B}(\mathbb{R}^d)$. Find $\int \mu(I_x)dx$

Question

Let $\mu$ be a probability measure on $\mathcal{B}(\mathbb{R}^d)$. Find $\int \mu(I_x)dx$, where

$$I_x := [x_1, x_1 + a_1]\times ... \times [x_d, x_d + a_d], a_j > 0$$

I have $\mu(A_1 \times ... \times A_d) = \mu_1(A_1)...\mu_d(A_d)$ for a and ll $A_i \in \mathscr{F}_i$ and $$\int f d\mu = \int ...\int f(x_1,...x_d)d\mu_1(x_1)...d\mu_d(x_d)$$ from the book, but I'm not really sure how to combine them to answer the question.

Answer 1

By Fubini's Theorem we have \begin{align} \int \mu(I_x) \mathrm{d} x &= \int \int 1_{[0,a_1] \times \ldots \times [0,a_d]} (w-x)d\mu^d(\omega_1,\ldots \omega_d) dx \\ &= \int \int 1_{[0,a_1] \times \ldots \times [0,a_d]} (w-x) dx \, d\mu^d(\omega_1,\ldots \omega_d) \\ &= a_1 \ldots a_d \mu(\mathbb{R})^d \\ &= a_1 \ldots a_d. \end{align}