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 $S$?
from Hot Weekly Questions - Mathematics Stack Exchange
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