  Fix $d\geq 2$ and $s\in (0,d)$. In this paper we introduce a notion called
small local action associated to a singular integral operator, which is a
necessary condition for the existence of principal value integral to exist. Our
goal is to understand the geometric properties of a measure for which an
associated singular integral has small local action. We revisit Mattila's
theory of symmetric measures and relate, under the condition that the measure
has finite upper density, the existence of small local action to the cost of
transporting the measure to a collection of symmetric measures. As
applications, we obtain a soft proof of a theorem of Tolsa and Ruiz-de-Villa on
the non-existence of a measure with positive and finite upper density for which
the principal value integral associated with the $s$-Riesz transform exists if
$s\not\in \mathbb{Z}$. Furthermore, we provide a considerable generalization of
this theorem if $s\in (d-1,d)$.
