How can I show that P is a probability measure on (\omega,sigma(X)) and how do i find the distribution of X?

Question

I have already done problem 1, problem 2 references #1. I need help showing that P is a probability measure and how to find its distribution

Answer 1

Look up the definition of probability measure and check that $P$ satisfies it.

For the distribution of $X$, it suffices to write down what $P(X=1)$ and $P(X=2)$ equal.