  Given a suitably nested family $Z = \langle Z(m,k,\gamma) \rangle_{m,k \in
\mathbb N, \gamma >0}$ of Borel subsets of matrices, and associated Borel
measures and rate function, $\mu$, an entropy, $\chi^{\mu}(Z)$, is introduced
which generalizes the microstates free entropy in free probability theory.
Under weak regularity conditions there exists a finite tuple of operators $X$
in a tracial von Neumann algebra such that \begin{eqnarray*} \chi^{\mu}(X) &
\geq & \chi^{\mu}(X \cap Z)
  & = & \chi^{\mu}(Z)\\ \end{eqnarray*} where $X \cap Z = \langle
\Gamma(X;m,k,\gamma) \cap Z(m,k,\gamma) \rangle_{m, k \in \mathbb N, \gamma
>0}$. This observation can be used to establish the existence of finite tuples
of operators with finite $\chi^{\mu}$-entropy. The intuition and proof come
from the bootstrap in statistical inference.
