  We consider the measure-geometric Laplacians $\Delta^{\mu}$ with respect to
atomless compactly supported Borel probability measures $\mu$ as introduced by
Freiberg and Z\"ahle in 2002 and show that the harmonic calculus of
$\Delta^{\mu}$ can be deduced from the classical (weak) Laplacian. We
explicitly calculate the eigenvalues and eigenfunctions of $\Delta^{\mu}$.
Further, it is shown that there exists a measure-geometric Laplacian whose
eigenfunctions are the Chebyshev polynomials and illustrate our results through
specific examples of fractal measures, namely Salem and inhomogeneous
self-similar Cantor measures.
