Let be an indexed collection of nonempty sets, and the coordinate maps. If is a -algebra on for each , the product -algebra on is the -algebra generated by

.

Denote this -algebra by . (If we also write or .)

**Proposition 1.** If is countable then is the -algebra generated by .

Proof. If , then where for ; on the other hand, .

**Proposition 2. **Suppose that $latex\mathcal M_\alpha$ is generated by , . Then is generated by . If is countable and for all , is generated by .

Proof. It’s easily seen that . On the other hand, for each , the collection is easily seen to be a -algebra on that contains and hence $latex\mathcal M_\alpha$.

**Proposition 3.** Let be metric spaces and let , equipped with the product metric. Then . If the ‘s are separable, then $latex\bigotimes_1^n\mathcal B_{X_j}=\mathcal B_X$.

Proof. By the Proposition 2, is generated by the sets , , where is open in . Since these sets are open in , Lemma implies that . Suppose now that is a countable dense set in , and let be the collection of balls in with rational radius and center in . Then every open set in is a union of members of (a countable union, since itself is countable). Moreover, the set of points in whose th coordinate is in for all is a countable dense subset of , and the balls of radius in are merely products of balls of radius in the ‘s. It follows that is generated by and is generated by . Therefore .

**Corollary. **.

**Definition.** An elementary family is a collection of subsets of such that:

1) ,

2) if then ,

3) if then is a finite disjoint union of members of .

**Proposition.** If is an elementary family, the collection of finite disjoint unions of members of is an algebra.

Proof. If and (, disjoint), then and , where these unions are disjoint, so and . It now follows by induction that if , then ; By inductive hypothesis we may assume that are disjoint, and then , which is a disjoint union. Suppose and with disjoint members of $latex\mathcal E$, then is closed under complements.

“The power of imagination is incredible. Often we see athletes achieving unbelievable results and wonder how they did it. One of the tools they use is visualization or mental imagery… they made the choice to create their destinies and visualized their achievements before they ultimately succeeded.” ~ George Kohlrieser

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 22-24.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.