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Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure ${\mu }_{x**}$ on the Cantor set. We use the measures ${\mu }_{x**}:x**\in ℬ₁\left(S¹\right)$ to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if ${\mu }_{T**(x**)}:x**\in ℬ₁\left(S¹\right)$ is non-separable as a subset of $ℳ\left(2^{\mathbb{N}}\right)$.