56 1. Measure theory From the definition of the lower Lebesgue integral, we conclude that λm({x Rd : f(x) λ}) Rd f(x) dx, and the claim follows. By sending λ to infinity or to zero, we obtain the following important corollary: Exercise 1.3.18. Let f : Rd [0, +∞] be measurable. (i) Show that if Rd f(x) dx ∞, then f is finite almost everywhere. Give a counterexample to show that the converse statement is false. (ii) Show that Rd f(x) dx = 0 if and only if f is zero almost every- where. Remark 1.3.16. The use of the integral Rd f(x) dx to control the distri- bution of f is known as the first moment method. One can also control this distribution using higher moments such as Rd |f(x)|p dx for various values of p, or exponential moments such as Rd etf(x) dx or the Fourier moments Rd eitf(x) dx for various values of t such moment methods are fundamental to probability theory. 1.3.4. Absolute integrability. Having set out the theory of the unsigned Lebesgue integral, we can now define the absolutely convergent Lebesgue integral. Definition 1.3.17 (Absolute integrability). An almost everywhere defined measurable function f : Rd C is said to be absolutely integrable if the unsigned integral f L1(Rd) := Rd |f(x)| dx is finite. We refer to this quantity f L1(Rd) as the L1(Rd) norm of f, and use L1(Rd) or L1(Rd C) to denote the space of absolutely inte- grable functions. If f is real-valued and absolutely integrable, we define the Lebesgue integral Rd f(x) dx by the formula (1.12) Rd f(x) dx := Rd f+(x) dx Rd f−(x) dx where f+ := max(f, 0), f− := max(−f, 0) are the magnitudes of the positive and negative components of f (note that the two unsigned integrals on the right-hand side are finite, as f+,f− are pointwise dominated by |f|). If f is complex-valued and absolutely integrable, we define the Lebesgue integral Rd f(x) dx by the formula Rd f(x) dx := Rd Re f(x) dx + i Rd Im f(x) dx
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