If we have a complex measurable , where are real and measurable, then we define the Lebesgue integral of over measurable set as follows:
where for a real function , and denote and respectively so that . We say that , that is, is Lebesgue-integrable, if . It is easy to verify the usual additive and scalar multiplicative properties of the complex Lebesgue integral.
Now that we can work with arbitrary complex measurable functions, we get the following inequality:
Result 12: .
Proof: This makes intuitive sense as it is essentially a triangle inequality. Think of as a complex number , and let be a “straightening rotation factor” so that and ; in other words, is simply rotated onto the real line. Then
Note, however, that these values are all real, so , and we are done. We will now use this triangle inequality to prove a result that gives sufficient conditions for the Lebesgue-integrability of the limit of a sequence of complex, measurable functions and further states that the integrals of those functions approach the integral of their limit.
Result 13: Given complex measurable functions which converge to some . Say they are “dominated” by some Lebesgue-integrable so that $|f_n|\le g$ over the entire space for all . Then is Lebesgue-integrable and .
Proof: We have essentially already proven that is measurable, and because is bounded above by , is certainly Lebesgue-integrable. To prove the second result, it suffices by Result 12 to prove that . Consider the distance between and . This is bounded above by , so consider . Fatou’s Lemma tells us that
Because approaches , the left side is . We can split up the right side into , so . But the limit is at most equal to the upper limit, and certainly , so we are done.