Union of Two Sigma-Algebras

Question

Question:

Let X be a non-empty set and let P and Q be two sigma-algebras on X. Is P U Q(union of P and Q) a sigma-algebra on X?

My approach towards the question:

My intuition was that this is not true in general. So, the next idea that came to my mind is to disaprove this by using an counterexample.

As the first step, I selected a non-empty set X as follows:

X = {a, b, c}, where a, b and c are distinct.

Then I listed out all the possible sigma-algebras that came to mind, which can be derived from X, as follows:

Here, @ denote the null set.

P1 = {@, X}

P2 = { @, X, {a}, {b, c} }

P3 = { @, X, {b}, {a, c} }

P4 = { @, X, {c}, {a, b} }

P5 = { @, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} }

Then, I considered P2 U P3 = { @, X, {a, b}, {a, c}, {b, c} }

So, in this case, {c}, the complement of {a, b} is not included in P2 U P3, hence not agreeing with the second property of a sigma-algebra.

Is this a valid counterexample which can prove that the intial statement given in the question is not valid in general.

I would appreciate any feedback on this.

Answer 1

Yes, your counterexample is valid and shows that the initial statement given in the question is not always true. By explicitly constructing the sigma-algebras P and Q and showing that their union P U Q does not satisfy the properties of a sigma-algebra, you have demonstrated that the claim is false in general. Specifically, your example shows that the union of two sigma-algebras need not be closed under complementation, which is the property that fails in this case.