1

I am studying the proof of the monotone class theorem in the book "Real analysis for graduate students" by Richard Bass, but I got stuck at a particular point. The theorem

The idea behind the proof is clear to me, we want to define subsets ($\mathcal{N}_i)$ of $\mathcal{M}$ that have properties of $\sigma$-Algebras and then show that these subsets ($\mathcal{N}_i)$ are actually the same as $\mathcal{M}$. This would show that the smallest monotone class that contains $A_0$ is in fact a $\sigma$-Algebra.

Now here is where I get stuck. I do not understand the claim I underlined in red. Part of the proof

$A_0$ is clearly a subset of $\mathcal{M}$. For each element $B_i$ in $A_0$, we know that $B_i^c$ is also in $A_0$, as $A_0$ is an algebra. But this is only for the elements IN $A_0$. How do we show that $A_0^c$ is an element in $\mathcal{M}$? (as this would imply that $A_0$ is in $\mathcal{N_1}$ by definition.)

Asaf Karagila
  • 393,674
UpzYaDead
  • 107
  • So $A_0$ is not a $\sigma -$algabra but an algebra only. Now, if $A\in A_0\subset \mathcal M$, as you said, $A^c\in A_0\subset \mathcal M$. Therefore $\mathcal N_1\supset A_0$. – Surb Aug 25 '21 at 14:48
  • I am not sure I understand the implication. The algebra $A_0$ itself is closed under complement and is contained in $\mathcal{M}$. But don't we have to show that the complement of the algebra $A_0$ is contained in $\mathcal{M}$? – UpzYaDead Aug 25 '21 at 16:44
  • What is Bass' definition of a monotone class? – DanielWainfleet Aug 25 '21 at 17:12
  • @DanielWainfleet, a monotone class $\mathcal{M}$ has 2 properties.

    1. If a sequence of sets $A_i\textuparrow A$ and each $A_i\in \mathcal{M}$, then $A\in\mathcal{M}$.

    2. If a sequence of sets $A_i\textdownarrow A$ and each $A_i\in \mathcal{M}$, then $A\in\mathcal{M}$.

    I do not see what you are trying to make figure out, it is probably something very trivial but I simply do not understand the implication as explained in my previous comment on Surb

     – UpzYaDead Aug 31 '21 at 15:09

0 Answers0