To make sure I don’t forget too much of what I’ve learnt in the past I periodically try solving some standard textbook exercises. But today I ran into a bit of an odd one in Rosenthal’s “A First Look at Rigorous Probability Theory”. Ex 2.7.4 (a) asks, “Let F_1, F_2, … be a sequence of collections of subsets of Omega, such that F_n is a subset of F_{n+1} for each n.

(a) Suppose that each F_i is an algebra. Prove that U_{i=1}^{inf} F_i is also an algebra.”

Now the given solution is trivially able to demonstrate that this infinite union of nested sets exhibits the defining properties of an algebra (in particular, closed under finite unions and complements) after making the following observation: **if A is in U_{i=1}^{inf} F_i then A is in F_i for ***some* i. But is this really correct? Imagine F_i to be the power set of 0,1,2,3,…,i plus the finite collection of finite unions between this power set and the complements of its members in the set of all integers (our Omega in this example). Then surely these F_i are all nested algebras? And, moreover, the set of all natural numbers is in U_{i=1}^{inf} F_i (but not in any particular F_i)?

Am I right, or have I forgotten something important??

(Another odd thing is that the solution given to part b uses a similar example to the counterexample I propose above.)

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