This article presents a comprehensive study of integration theory for complex-valued measurable functions with respect to a positive measure. The main objective is to establish the necessary and sufficient conditions for the integrability of complex-valued measurable functions and to determine the algebraic structure of the set of these integrable functions. We prove that a measurable function f:X?Cis integrable on R E?Aif and only if both its real Re(f)and imaginary parts Im(f)are integrable on R E. Furthermore, we prove that the set L(X,A,µ,C)of complex-valued integrable functions forms a vector space on R Cequipped with the usual operations of addition and scalar multiplication. Several fundamental properties are established, including linearity, the relationship between the integrability of R fand that of R |f|, and the behavior of the integral R on sets of measure zero. This work extends the classical results of real integration theory to the complex setting and provides a rigorous foundation for applications in analysis, probability theory, and mathematical physics.
Measurable functions, complex-valued functions, positive measure, Lebesgue integral, vector space, integrability conditions, real and imaginary parts, measure theory.
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