Probability with Measure
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4.8 Lebesgue integration of complex valued functions \((\Delta )\)
The definition of the Lebesgue integral can be extended to complex valued functions. Let \((S, \Sigma , m)\) be a measure space and \(f:S \rightarrow \C \). We can always write \(f = f_1 + if_2\), where the real and imaginary parts are \(f_{i}:S \rightarrow \R ~(i=1,2)\). If both \(f_1\) and \(f_2\) are measurable then we say that \(f\) is measurable1. If both \(f_1\) and \(f_2\) have integrals according to Definition 4.5.1 then we define
\[ \int _{S}f \,dm = \int _{S}f_{1}\,dm + i\int _{S}f_{2} \,dm.\]
Recall that for \(z=x+iy\in \C \), where \(x\) and \(y\) are the real and imaginary parts of \(f\), we define \(|z|=(x^2+y^2)^{1/2}\). Correspondingly, we make the pointwise definition that when \(f:S\to \C \), \(|f|=(f_1+f_2)^{1/2}\). Note that \(|f|:S\to \R \). This allows us to define a complex version of \(\Lone \), given by
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