# CHAPTER 10

# PRODUCT MEASURES

Product measures are measures defined on the product of several measurable spaces. Fubini’s theorem gives conditions under which we can evaluate integrals with respect to product measures by computing iterated integrals. In this chapter, we present the definition of product measures and Fubini’s theorem.

# 10.1 Basic Concepts and Facts

**Definition 10.1** (Cartesian Product). Let *S*_{1} and *S*_{2} be two sets. The Cartesian product of *S*_{1} and *S*_{2} is a set defined as

**Definition 10.2** (Measurable Rectangle). Let (*S*_{1}, ∑_{1}) and (*S*_{2}, ∑_{2}) be two measurable spaces. Let *S* = *S*_{1} × *S*_{2} be the Cartesian product of *S*_{1} and *S*_{2}. A measurable rectangle in the product space *S* is a product *A*_{1} × *A*_{2} for which *A*_{1} ∑_{1} and *A*_{2} ∑_{2}.

**Definition 10.3** (Product σ-Algebra). Let (*S*_{1}, ∑_{1}) and (*S*_{2}, ∑_{2}) be two measurable spaces. The product σ-algebra ∑ ∑_{2} (∑_{1} ∑_{2} is not a Cartesian product in the usual sense.) is defined as the σ-algebra generated by all measurable rectangles in the product space *S*_{1} × *S*_{2}:

**Definition ...**

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