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This study addresses the often-overlooked issue of measurability atintermediate points when applying Taylor's theorems to random functions andrandom vectors (e.g., likelihood functions with respect to estimators) instatistics. Classical Taylor-related theorems were originally developed fordeterministic settings. Consequently, they do not directly extend to stochasticfunctions and variables and do not inherently guarantee the measurability ofintermediate points. In statistical contexts, applying these theorems withoutproperly accounting for randomness can lead to analyses that lack well-definedprobabilistic interpretations. Elementary approaches, such as pointwiseconstructions, are insufficient for handling random quantities and establishingmeasurable intermediate points. Moreover, some statistical literature hasimplicitly disregarded this issue, often neglecting the stochastic nature ofthe problem and assuming that intermediate points are measurable. To addressthis gap, we develop multivariate Taylor's and mean value theorems tailored forrandom functions and random variables under mild assumptions. We provideillustrative examples demonstrating the applicability of our results tocommonly used statistical methods, including maximum likelihood estimation,$M$-estimation, and profile estimation. Our findings contribute a rigorousfoundation for the applications of Taylor expansions in statistics.

Taylor's Theorem and Mean Value Theorem for Random Functions and Random Variables