Is the following probability statement about random variables true?

Question

Let $X: \Omega \to \mathbb{R}$ be a random variable.

Suppose that $A \in \mathcal{R}$, the Borel sets of the real numbers and for every $a \in A$ we have $\mathbb{P}\{X=a\} = 0$. Is it true that $\mathbb{P}\{X \in A \} = 0$?

I have trouble proving this due to the possible uncountability, so I am thinking it might be false. Any help is appreciated.

Answer 1

No.

If $X$ has continuous distribution (i.e. its CDF is continuous) then $\mathsf P(X=x)=0$ for every $x\in\mathbb R$.

Nevertheless $\mathsf P(X\in\mathbb R)=1$

Answer 2

Hint: Consider a random variable that is uniformly distributed in $A=[0,1]$.