# The Riesz Representation Theorem for complex functions

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?

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 $$\int_X f \; d\mu = \int_X u \; d\mu + i \int_X v \; d\mu$$.