  We develop new local $T1$ theorems to characterize Calder\'on-Zygmund
operators that extend boundedly or compactly on $L^{p}(\mathbb R^{n},\mu)$ with
$\mu$ a measure of power growth.
  The results, whose proofs do not require random grids, allow the use of a
countable collection of testing functions.
  As a corollary, we describe the measures $\mu$ of the complex plane for which
the Cauchy integral defines a compact operator on $L^p(\mathbb C,\mu)$.
