Introduction to Measure theory

Question

I just started learning measure theory and I found the following question in A First Look at Rigorous Probability by Rosenthal. Given are $\mathcal J = \{ \text{intervals in } \Omega = [0,1] \}$ and $\mathcal B_0 = \{ \text{all finite unions of elements from } \mathcal J \}$. We are asked to prove that $\mathcal B_0$ is an algebra. I managed to establish that $$(i) \ \emptyset \in \mathcal B_0,$$ $$(ii) \ \Omega \in \mathcal B_0,$$ $$(iii) \ B_i \in \mathcal B_0 \Rightarrow \bigcup_{i=1}^{k_1} B_i = \bigcup_{i=1}^{k_1} \left( \bigcup_{j=1}^{k_2} A_{ij} \right) = \bigcup_{i=1}^{k_1k_2} A_i \in \mathcal B_0,$$ but I am having trouble both with the formation of complement and finite intersection part. I was wondering if anybody was willing to give me a hint so I can proceed with the question.

Answer 1

To show closure under finite intersections, use the fact that intersections distribute over unions; in other words, for any families of sets $\{A_i\}$, $\{B_j\}$, we have the equality of sets $$\left(\bigcup_i A_i\right)\cap\left(\bigcup_j B_j\right) = \bigcup_{i,j} (A_i\cap B_j).$$

Once you have shown closure under finite intersections, closure under complements follows (why?) from De Morgan’s laws and the fact that the complements of intervals lie in $\mathcal{B}_0$.