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Gaussian processes appear as building blocks in various stochastic models andhave been found instrumental to account for imprecisely known, latentfunctions. It is often the case that such functions may be directly orindirectly evaluated, be it in static or in sequential settings. Here we focuson situations where, rather than pointwise evaluations, evaluations ofprescribed linear operators at the function of interest are (sequentially)assimilated. While working with operator data is increasingly encountered inthe practice of Gaussian process modelling, mathematical details ofconditioning and model updating in such settings are typically by-passed. Herewe address these questions by highlighting conditions under which Gaussianprocess modelling coincides with endowing separable Banach spaces of functionswith Gaussian measures, and by leveraging existing results on thedisintegration of such measures with respect to operator data. Using recentresults on path properties of GPs and their connection to RKHS, we extend theGaussian process - Gaussian measure correspondence beyond the standard settingof Gaussian random elements in the Banach space of continuous functions.Turning then to the sequential settings, we revisit update formulae in theGaussian measure framework and establish equalities between final andintermediate posterior mean functions and covariance operators. The latterequalities appear as infinite-dimensional and discretization-independentanalogues of Gaussian vector update formulae.

Disintegration of Gaussian Measures for Sequential Assimilation of Linear Operator Data