Exercise 2.19 (Independence of constant random variables)

Let $X$ be our deterministic random variable, and deduce its range from $X:(\mathrm{\Omega },F)\to (R_1,B_1)$. Any deterministic random variable satisfies

for some $c\in R_1$. Let $Y$ be another random variable with the range in $(R_2,B_2)$. Then, for any $S_1\in B_1,S_2\in B_2$, we have two cases:

• $c\in S_1$. We then have, $P(X\in S_1,Y\in S_2)=P(Y\in S_2)$, which is exactly the product that we want, since $P(X\in S_1)=1$.
• $c\notin S_1$. We then have, $P(\varnothing )=P(X\in S_1,Y\in S_2)=0$, which is exactly the product that we want, since $P(X\in S_1)=0$.