Let $\mathcal B$ denote a sigma field associated with a sample space $\omega$. Suppose $A$ and $B$ are members of $\mathcal B$. Let $P$ be a probability function defined on $\mathcal B$. Show that $$ P(A) + P(B) \le 2 (P(A)P(B))^\frac 12 $$
I am stuck on how to prove this. I tried using the Bonferroni's Inequality. But that didnt help me much. How do i approach this?
Assertion cannot be true for arbitrary $A,B$.
$P(A) + P(B) \leq 2(P(A)P(B))^{1/2} \iff (P(A) + P(B))^2 \leq 4P(A)P(B) \iff P(A)^2 -2P(A)P(B) + P(B)^2 = (P(A) -P(B))^2 \leq 0 \iff P(A) = P(B)$
Hence, it is true iff $P(A) = P(B)$, in which case the condition is reduced to a trivial statement.