# Relation between the measures of two sets defined via Lebesgue integration

Suppose $$a : \mathbb R_+ \to \{-1,1\}$$ is a measurable function. Let $$X_0 =\frac12$$. Consider a particle that moves on the $$X-$$axis as follows. $$X_t = X_0 + \int_0^t a_s ds$$ where the integral is a Lebesgue integral.

Fix a $$T=\frac12$$. So, $$X_t \in [0,1]$$ for all $$t \le T$$.

Let $$S \subset [0,1]$$ be a set such that $$\ell(S) =1$$, where $$\ell(\cdot)$$ is the Lebesgue measure.

Define, $$G:= \{t \le T: X_t \in S\}.$$

Is it the case that $$\ell(G) = \ell([0,T]) = \frac12$$?

That is, the particle spends almost no time outside $$S$$?