Given a
measurable space , an extended real-valued function is
measurable if and only if its positive and negative parts are. Therefore, if such a function is measurable, so is its absolute value , being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking as f = 1_V - \frac{1}{2}, where is a
Vitali set, it is clear that is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the
Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a
signed measure into positive and negative parts — see the
Hahn decomposition theorem. ==See also==