A measure which takes values in the complex numbers. The set of complex measures on a measure space X forms a vector space. Note that this is not the case for the more common positive measures. Also, the space of finite measures ( left bracketing bar μ(X) right bracketing bar <∞) has a norm given by the total variation measure left double bracketing bar μ right double bracketing bar = left bracketing bar μ right bracketing bar (X)|, which makes it a Banach space. Using the polar representation of μ, it is possible to define the Lebesgue integral using a complex measure, integral f d μ = integral e^(i θ) f d left bracketing bar μ right bracketing bar .

Banach space | Lebesgue integral | measure | measure space | polar representation