Probability of X ∈ [0, 1/2) for X(ω) = ω^2 on [0, 1] with Lebesgue Measure
P(X ∈ [0, 1/2)) = P(ω^2 ∈ [0, 1/2)) = P(ω ∈ [0, √1/2])
= ∫[0, √1/2] 1 dω = ∫[0, √1/2] dω = ω|[0, √1/2] = (√1/2 - 0) = √1/2
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