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In measure theory, a branch of mathematical analysis, a function is said to be **𝒞-almeasurable** (or more commonly referred to as **measurable**) when it preserves the measurable structure of the sets it operates on. The concept is fundamental to integration theory and probability, where measurable functions allow the extension of integration from simple functions to more complex ones.

A function *f*: *X* → *Y* between measurable spaces (*X*, 𝒞ₓ) and (*Y*, 𝒞ᵧ) is called 𝒞-almeasurable if the preimage of every set in 𝒞ᵧ is a measurable set in 𝒞ₓ. Formally, for every *A* ∈ 𝒞ᵧ, the set *f⁻¹*(*A*) = {*x* ∈ *X* | *f*(*x*) ∈ *A*} must belong to 𝒞ₓ. This property ensures that measurable functions can be integrated with respect to a measure defined on 𝒞ₓ.

Measurable functions are crucial in constructing the Lebesgue integral, which generalizes the Riemann integral. They also play a key role in probability theory, where random variables are typically required to be measurable with respect to the σ-algebra of events. The class of measurable functions is closed under algebraic operations, limits, and composition with other measurable functions, making them a well-behaved class for analysis.

In practice, verifying measurability often relies on simpler conditions, such as continuity (for Borel measurable functions) or the fact that monotone functions are measurable. The concept extends naturally to vector-valued functions and functions between more general measurable spaces, forming the basis for advanced topics in real analysis and probability.