  Let $S^1$ be the stopping time space and $\mathcal{B}_1(S^1)$ be the Baire-1
elements of the second dual of $S^1$. To each element $x^{**}$ in the space
$\mathcal{B}_1(S^1)$ we associate a positive Borel measure $\mu_{x^{**}}$ on
the Cantor set. We use the measures $\{\mu_{x^{**}}:
x^{**}\in\mathcal{B}_1(S^1) \}$ to characterize the operators $T:X\to S^1$,
defined on a space $X$ with an unconditional basis, which preserve a copy of
$S^1$. In particular, we show that $T$ preserves a copy of $S^1$ if and only if
the set $\{\mu_{x^{**}}:\;x^{**}\in\mathcal{B}_1(S^1)\}$ is non separable as a
subset of $\mathcal{M}(2^\mathbb{N})$.
