  We introduce a novel approach for defining a $\delta'$-interaction on a
subset of the real line of Lebesgue measure zero which is based on
Sturm-Liouville differential expression with measure coefficients. This enables
us to establish basic spectral properties (e.g., self-adjointness, lower
semiboundedness and spectral asymptotics) of Hamiltonians with
$\delta'$-interactions concentrated on sets of complicated structures.
