This thesis presents a general algebraic approach for indirectly measuring both classical and quantum observables, along with several applications. To handle the case of imperfectly correlated indirect detectors we generalize the observable spectra from eigenvalues to contextual values. Eigenvalues weight spectral idempotents to construct an observable, but contextual values can weight more general probability observables corresponding to indirect detector outcomes in order to construct the same observable. We develop the classical case using the logical approach of Bayesian probability theory to emphasize the generality of the concept. For the quantum case, we outline how to generalize the classical case in a straightforward manner by treating the classical sample space as a spectral idempotent decomposition of the enveloping algebra for a Lie group; such a sample space can then be rotated to other equivalent sample spaces through Lie group automorphisms. We give several classical and quantum examples to illustrate the utility of our approach. In particular, we use the approach to describe the theoretical derivation and experimental violation of generalized Leggett-Garg inequalities using a quantum optical setup. We also describe the measurement of which-path information using an electronic Mach-Zehnder interferometer. Finally, we provide a detailed and exact treatment of the quantum weak value, which appears as a general feature in conditioned observable measurements using a weakly correlated detector.