Thread: Measurability of of Random Variables w.r.t. Sigma Fields

1. Measurability of of Random Variables w.r.t. Sigma Fields

Hello everyone,

I'm having a little trouble with a probability problem with three parts; I think I'm having trouble wrapping my head around just what's going on here. If anyone could give me a starting point, I'd appreciate it.

Here's the problem (Billingsley 5.1) (X a random variable)

a. Show that X is measurable w.r.t. the sigma field J iff sigma(X) is a subset of J. Show that X is a measurable w.r.t. sigma(Y) iff sigma(x) is a subset of sigma(Y)

b. Show that if J = {empty set, omega}, then X is measurable w.r.t. J iff X is constant.

c. Suppose that P(A) is 0 or 1 for every A in J. This holds, for example, if J is the tail field of an independent sequence, or if J consists of the countable and cocountable sets on the unit interval with Lebesgue measure. Show that if X is measurable w.r.t. J, then P[X=c] = 1 for some constant c.

Thanks for any and all help!

Best regards

2. Re: Measurability of of Random Variables w.r.t. Sigma Fields

For the first question, use the fact that $\sigma (X)$ is a $\sigma$-algebra which consists of the elements of the form $X^{-1}(B)$ where $B$ is in the Borel $\sigma$-algebra.
For the second question, if $X$ is measurable w.r.t. $J$ then for each $r\in\mathbb R$ we have $X^{-1}(\{r\})=\emptyset$ or $X$. There is only one $r$ such that $X^{-1}(\{r\})=X$.
For the third question, what about $\inf\{t\in\mathbb R: P(X\leq t)=1\}$?

3. Re: Measurability of of Random Variables w.r.t. Sigma Fields

Thank you so much! You were a big help.