Mérhetsége

Mérhetsége refers to the mathematical property of being measurable with respect to a given measure or σ-algebra. In measure theory, a set, function, or object is described as measurable if its structure is compatible with the underlying measure, allowing the assignment of sizes, probabilities, or integrals in a consistent way.

For sets, a subset E of a space X is measurable if it belongs to the σ-algebra considered on X. In the common real-number setting, this often means E is Lebesgue measurable with respect to the Lebesgue σ-algebra, or Borel measurable with respect to the Borel σ-algebra. For a function, measurability is defined through preimages: a function f: X → ℝ is measurable if, for every Borel set B ⊆ ℝ, the preimage f⁻¹(B) is a measurable subset of X. This requirement guarantees that the function interacts well with the measure and can be integrated.

Key examples include all open, closed, and countable sets being measurable in standard real analysis, and any continuous function being measurable with respect to the Borel σ-algebra. Simple functions and indicator functions of measurable sets are foundational in constructing more complex measurable objects.

Variants of mérhetsége arise with different measures and σ-algebras. Lebesgue measurability is a central concept on the real line, while Borel measurability concerns the σ-algebra generated by open sets. Measurability is crucial in probability theory, where random variables are modeled as measurable functions, enabling the definition of expectations and distributions. Non-measurable sets can exist under certain axioms of choice, illustrating the dependence of mérhetsége on the chosen mathematical framework.