Probability of X ∈ [0, 1/2) for X(ω) = ω^2 on Ω = [0, 1]

Let Ω = [0, 1] with Borel sets and Lebesgue measure. Find P(X ∈ [0, 1/2)) for X(ω) = ω^2.

P(X ∈ [0, 1/2)) = P(ω^2 ∈ [0, 1/2))

P(ω^2 ∈ [0, 1/2)) = P(ω ∈ [0, √1/2)) = √1/2 - 0 = √1/2