ölçülebilen

ölçülebilen is a Turkish term meaning measurable, used primarily in mathematics to describe objects that are compatible with a given measure. In a measure space (X, F, μ), a subset A ⊆ X is called measurable (ölçülebilen) if A belongs to the σ-algebra F that defines the measurement on X. The collection of all measurable sets forms a σ-algebra, providing the framework for defining and manipulating sizes and probabilities.

For functions, a function f: X → ℝ is said to be measurable (ölçülebilen) with respect to F if the preimage of every Borel set in ℝ is in F. Equivalently, for every real number α, the set {x ∈ X : f(x) > α} (and similarly {f ≤ α}, {f < α}, {f ≥ α}) is measurable. This notion is central to integration and probability because it ensures that one can assign meaningful integrals and expectations to f.

Examples: All Borel sets in the real line are Lebesgue measurable; the indicator function of a measurable set A is measurable; simple functions, which are finite linear combinations of indicator functions of measurable sets, are measurable; the pointwise limit of a sequence of measurable functions is measurable under standard conditions.

Notes: There exist non-measurable sets, depending on the chosen measure and the axioms of set theory; measurability is relative to the given measure structure. The term also appears as ölçülebilen küme (measurable set) and ölçülebilen fonksiyon (measurable function).

Applications: Measurability underpins probability theory, where random variables are modeled as measurable functions, and integration theory, where the integral is defined for measurable functions.