  The notion of a coalgebra measuring, introduced by Sweedler, is a kind of
generalized ring map between algebras. We begin by studying maps on Hochschild
homology induced by coalgebra measurings. We then introduce a notion of
coalgebra measuring between Lie algebras and use it to obtain maps on Lie
algebra homology. Further, these measurings between Lie algebras satisfy nice
adjoint like properties with respect to universal enveloping algebras.
  More generally, we introduce and undertake a detailed study of the notion of
coalgebra measuring between algebras over any operad $\mathcal O$. In case
$\mathcal O$ is a binary and quadratic operad, we show that a measuring of
$\mathcal O$-algebras leads to maps on operadic homology. In general, for any
operad $\mathcal O$, we construct universal measuring coalgebras to show that
the category of $\mathcal O$-algebras is enriched over coalgebras. We develop
measuring comodules and universal measuring comodules for this theory. We also
relate these to measurings of the universal enveloping algebra $U_{\mathcal
O}(\mathscr A)$ of an $\mathcal O$-algebra $\mathscr A$ and the modules over
it. Finally, we construct the Sweedler product $C\rhd \mathscr A$ of a
coalgebra $C$ and an $\mathcal O$-algebra $\mathscr A$. The object $C\rhd
\mathscr A$ is universal among $\mathcal O$-algebras that arise as targets of
$C$-measurings starting from $\mathscr A$.
