Complex measure

In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. Formally, a complex measure μ {displaystyle mu } on a measurable space ( X , Σ ) {displaystyle (X,Sigma )} is a complex-valued function that is sigma-additive. In other words, for any sequence ( A n ) n ∈ N {displaystyle (A_{n})_{nin mathbb {N} }} of disjoint sets belonging to Σ {displaystyle Sigma } , one has As ⋃ n = 1 ∞ A n = ⋃ n = 1 ∞ A σ ( n ) {displaystyle displaystyle igcup _{n=1}^{infty }A_{n}=igcup _{n=1}^{infty }A_{sigma (n)}} for any permutation σ : N → N {displaystyle sigma :mathbb {N} o mathbb {N} } , it follows that ∑ n = 1 ∞ μ ( A n ) {displaystyle displaystyle sum _{n=1}^{infty }mu (A_{n})} converges unconditionally (hence absolutely). One can define the integral of a complex-valued measurable function with respect to a complex measure in the same way as the Lebesgue integral of a real-valued measurable function with respect to a non-negative measure, by approximating a measurable function with simple functions. Just as in the case of ordinary integration, this more general integral might fail to exist, or its value might be infinite (the complex infinity). Another approach is to not develop a theory of integration from scratch, but rather use the already available concept of integral of a real-valued function with respect to a non-negative measure. To that end, it is a quick check that the real and imaginary parts μ1 and μ2 of a complex measure μ are finite-valued signed measures. One can apply the Hahn-Jordan decomposition to these measures to split them as