  We show that a Krein-Feller operator is naturally associated to a fixed
measure $\mu$, assumed positive, $\sigma$-finite, and non-atomic. Dual pairs of
operators are introduced, carried by the two Hilbert spaces,
$L^{2}\left(\mu\right)$ and $L^{2}\left(\lambda\right)$, where $\lambda$
denotes Lebesgue measure. An associated operator pair consists of two specific
densely defined (unbounded) operators, each one contained in the adjoint of the
other. This then yields a rigorous analysis of the corresponding
$\mu$-Krein-Feller operator as a closable quadratic form. As an application,
for a given measure $\mu$, including the case of fractal measures, we compute
the associated diffusion, semigroup, Dirichlet forms, and $\mu$-generalized
heat equation.
