In mathematics, Banach measure in measure theory may mean a real-valued function on the algebra of all sets (for example, in the plane), by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round the phenomenon of non-measurable sets. However, as the Vitali set shows, it cannot be countably additive.
The existence of Banach measures proves the impossibility of a Banach–Tarski paradox in two dimensions.
The concept of Banach measure is to be distinguished from the idea of a measure taking values in a Banach space, for example in the theory of spectral measures.