  For a an arbitrary periodic Borel measure $\mu$, we prove order
$O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar
elliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with
$\varepsilon$-periodic coefficients, $\varepsilon>0.$ Here $\mu^\varepsilon$ is
the measure obtained by $\varepsilon$-scaling of $\mu.$ Our analysis includes
both the case of a measure absolutely continuous with respect to the standard
Lebesgue measure and the case of "singular" periodic structures (or
"multistructures"), when $\mu$ is supported by lower-dimensional manifolds.
