Lebesguemåttmätbara

Lebesguemåttmätbara refers to a concept in measure theory, specifically within the context of the Lebesgue measure. In this mathematical framework, a set is considered Lebesgue measuremätbar (Lebesgue measurable) if its "size" can be meaningfully defined by the Lebesgue measure. The Lebesgue measure is a generalization of length for intervals, area for rectangles, and volume for higher-dimensional objects. It assigns a non-negative real number to certain subsets of Euclidean space.

For a set to be Lebesgue measuremätbar, it must satisfy a specific condition related to how it interacts with any measurable subset of the space. This condition essentially ensures that the set does not have an "unmanageable" or "pathological" structure that would prevent a consistent assignment of measure. Not all subsets of Euclidean space are Lebesgue measuremätbar; constructing non-measurable sets requires the Axiom of Choice.

The property of being Lebesgue measuremätbar is crucial for the development of integration theory, particularly the Lebesgue integral. The Lebesgue integral allows for the integration of a much wider class of functions than the Riemann integral, and its power relies heavily on the notion of measurable sets. Functions that are constant on measurable sets, or whose preimages of intervals are measurable, are considered Lebesgue integrable if they are measurable themselves. The theory of Lebesgue measuremätbara sets forms a foundational pillar of modern analysis and probability theory.