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The Riesz Representation Theorem: Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_c(X)$ (the set of complex functions with compact support). Then there is a unique positive measure ${\mu}$ on $\mathcal{B}(X)$, the Borel $\sigma$-algebra associated with the topology on $X$, for which $$ I(f)= \int_X f \,d{\mu} \text{ for all } f \in C_c(f) $$

To prove this theorem, Walter Rudin (Real and Complex Analysis) says that it is sufficient to prove it for real functions $f$ on $X$.

My question is: why, if it holds for real functions, should it hold for complex functions?

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Because a complex-valued continuous function $f$ of compact support can be written as $f = u + i v$ where $u$ and $v$ are real-valued continuous functions of compact support; $I(f) = I(u) + i I(v)$ and $\int_X f \; d\mu = \int_X u \; d\mu + i \int_X v \; d\mu$.

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