5 Iterated integrals
Key takeaways:
- The product \(\sigma\)-algebra is characterized by the section property (proposition ??).
- \(\sigma\)-finiteness is extremely important.
Properties of products
Recall that the product \(\sigma\)-algebra \(\Sigma_1\times \Sigma_2\) is freely generated by measurable rectangles (definition 2.6).
Definition 5.1 ((cross) sections) Given sets \(X, Y\) and \(E\subset X\times Y\), define for \(x\in X, y\in Y\) the sections \[ [E]_x = \{y\in Y: (x, y)\in E\}, \quad [E]^y = \{x\in X: (x, y)\in E\} \] The notation should be reminiscent of slicing along \(X=x\) on the horizontal, or \(Y=y\) on the vertical axis. Similarly, the sections of \(f\) defined on \(X\times Y\) is \[ [f]_x(y) = [f]^y(x) = f(x, y) \]
Proposition 5.1 (sections of measurable sets are measurable) The product \(\sigma\)-algebra \(\Sigma = \Sigma_1\times \Sigma_2\) has the section property that, \(\forall A\in \Sigma\), \[ [A]_{\forall x\in \Omega_1}\in \Sigma_2, \quad [A]^{\forall y\in \Omega_2} \in \Sigma_1 \]
Proof
The set \(\Sigma'\) of all subsets of \(\Omega_1\times \Omega_2\) (regarded as Cartesian product of sets) satisfying the section property is a \(\sigma\)-algebra containing \(\Sigma_1\times \Sigma_2\) (since sections of the Cartesian product of measurable sets are measurable); the proof concludes by the universal property of \(\Sigma_1\times \Sigma_2\), which is the smallest \(\sigma\)-algebra generated by products of measurable sets.Proposition 5.2 (sections of measurable functions are measurable) Given a \(\Sigma_1\times \Sigma_2\) measurable function \(f\) on \(\Omega_1\times \Omega_2\), then \([f]_{\forall x\in \Omega_1}\) is a measurable function on \(\Omega_2\), and \([f]^{\forall y\in \Omega_2}\) is a measurable function on \(\Omega_1\).
Definition 5.2 (product measures) The product of two measure spaces is \[ (\Omega_1\times \Omega_2, \Sigma_1\times \Sigma_2, \mu_1\times \mu_2) = (\Omega_1, \Sigma_1, \mu_1)\times (\Omega_2, \Sigma_2, \mu_2) \] where \(\Omega_1\times \Omega_2\) is the Cartesian product of sets, \(\Sigma_1\times \Sigma_2\) is the \(\sigma\)-algebra product 2.6, and \(\mu_1\times \mu_2\) is uniquely defined by the requirement \[ \mu(A_1\times A_2) = \mu_1(A_1) \mu_2(A_2) \] The exact definition and uniqueness of the product measure is postponed to theorem 5.1.