Studying for a probability exam and here are two practice problems I can't seem to solve

1. Let P be the uniform distribution on a finite set Omega and let A be a subset of Omega. Prove that P(.|A) is the uniform distribution on A.

2. Let X and Y be random variables and let A be an event. Prove the function, Z(w) = X(w) if w is in A and Y(w) if w is not in A , is a random variable

thanks for any help getting me started

Both are theoretical questions. So it is very difficult to write here.
So I am not giving the full solution.
1. I guess the set A is an interval ( say [c,d] ) . Let P be the uniform distribution on the interval [a,b]
P(x) = 1/(b-a)
& P(A) = (d-c)/(b-a)

P(x/A) = P(x and A) / P(A)
= P(x)/P(A)
= 1/(b-a) / (d-c)/(b-a)
= 1/(d-c) it is a uniform distribution.

2) Z(w) is a real valued function. find out the probability space of Z.