0are

0are is a term used to denote a zero-area object in the plane or in higher-dimensional space. In mathematical contexts, it is convenient to refer to a set with zero two-dimensional Lebesgue measure as a 0are set. The concept is central to measure theory and geometric analysis, where area is treated as a measurable quantity that may vanish for certain sets.

Formally, in the Euclidean plane R^2, a subset A is called a 0are set if its Lebesgue measure is zero. This includes any line or smooth curve, the graph of a function, finite or countable point sets, and many other thin geometries. In higher dimensions, a similar notion applies: a subset of R^n with zero n-dimensional volume is a 0are set in that space. A key consequence is that a 0are set cannot contain any open region.

Basic properties follow from measure theory. The interior of a 0are set is empty. The class of 0are sets is closed under countable unions and subsets: the union of countably many 0are sets remains 0are, as Lebesgue measure is countably additive. Conversely, any set with nonempty interior has positive area in its ambient plane.

Applications of the concept appear in analysis and geometry when distinguishing negligible regions from those contributing to area calculations. It helps in discussions of integration, approximation, and the behavior of measures under mappings, where sets of zero area can sometimes be ignored.

See also: Lebesgue measure, measure theory, null set, fractal dimension, geometric measure theory.