- Are almost surely constant random variables trivial sigma-algebra-measurable?
These links suggest no:
This link suggests yes:
I really don't think a random variable $X(\omega) = 2 \ \forall \omega \in \Omega$ except for a set $A \in \mathscr F$ can ever be trivial sigma-algebra-measurable even if A has a probability of zero. Is Dr Mattingly (same link as earlier) wrong?
- What about the converse? Are trivial sigma-algebra-measurable random variables necessarily constant (that is never almost surely constant)?