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In materials informatics, the act of measuring a material property is routinely treated as a neutral act of passive observation. Yet, every measurement consumes finite resources, physically alters the sample, or reshapes the space of future measurements through model-guided selection. This paper identifies a direct analog of the quantum measurement problem within data-driven materials discovery: observation is not merely informative but constitutively changes the system being observed by depleting experimental budgets, inducing material modifications, and biasing the very distribution of data that subsequent AI models will learn. The theoretical claim advanced here is that materials informatics harbors an intrinsic measurement problem in which AI-guided measurement actively constructs rather than neutrally samples the observable landscape, thereby rendering the resulting datasets and models path-dependent on the history of prior observations. Key concepts include resource depletion, selection feedback loops, and measurement-driven evolution, all of which distinguish classical materials measurement effects from quantum collapse while sharing the core epistemic feature of non-neutrality. The implications are far-reaching for AI-guided materials discovery: autonomous laboratories must treat measurement policies as interventions rather than recordings, active-learning algorithms must internalize the cost of altering the observable world, and dataset curation protocols must document measurement history as rigorously as they document final property values. By theorizing this measurement problem, the present analysis offers a conceptual framework that reframes experiment design, model training, and discovery workflows as inherently self-referential processes in which the observer and the observed co-evolve.