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We show that a Krein-Feller operator is naturally associated to a fixedmeasure $\mu$, assumed positive, $\sigma$-finite, and non-atomic. Dual pairs ofoperators 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 specificdensely defined (unbounded) operators, each one contained in the adjoint of theother. 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 computethe associated diffusion, semigroup, Dirichlet forms, and $\mu$-generalizedheat equation.

Dual pairs of operators, harmonic analysis of singular non-atomic measures and Krein-Feller diffusion