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In mathematics, the positive part of a real or extended real-valued function is defined by the formula : Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the ''x''-axis, and letting take the value zero there. Similarly, the negative part of ''f'' is defined as : Note that both ''f''+ and ''f''− are non-negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function ''f'' can be expressed in terms of ''f''+ and ''f''− as : Also note that :. Using these two equations one may express the positive and negative parts as : : Another representation, using the Iverson bracket is : : One may define the positive and negative part of any function with values in a linearly ordered group. ==Measure-theoretic properties== Given a measurable space (''X'',Σ), an extended real-valued function ''f'' is measurable if and only if its positive and negative parts are. Therefore, if such a function ''f'' is measurable, so is its absolute value |''f''|, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking ''f'' as : where ''V'' is a Vitali set, it is clear that ''f'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Positive and negative parts」の詳細全文を読む スポンサード リンク
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