Is there anyone who is familar with Meausre Theory?

Question

If so, would appreciate your guidance on below problem.

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?

Answer 1

Yes, your approach and counterexample are correct. Your counterexample shows that the union of two sigma-algebras is not necessarily itself a sigma-algebra. In general, the union of two sigma-algebras is a sigma-algebra if and only if one of them is a subset of the other one. So, your counterexample shows that the initial statement given in the question is not always true.

Answer 2

Thank you!