1.3. The Lebesgue integral 57 where the two integrals on the right are interpreted as real-valued absolutely integrable Lebesgue integrals. It is easy to see that the unsigned, real- valued, and complex-valued Lebesgue integrals defined in this manner are compatible on their common domains of definition. Note from construction that the absolutely integrable Lebesgue integral extends the absolutely integrable simple integral, which is now redundant and will not be needed any further in the sequel. Remark 1.3.18. One can attempt to define integrals for non-absolutely- integrable functions, analogous to the improper integrals ∞ 0 f(x) dx := limR→∞ R 0 f(x) dx or the principal value integrals p.v. ∞ −∞ f(x) dx := limR→∞ R −R f(x) dx one sees in the classical one-dimensional Riemannian theory. While one can certainly generate any number of such extensions of the Lebesgue integral concept, such extensions tend to be poorly behaved with respect to various important operations, such as change of variables or exchanging limits and integrals, so it is usually not worthwhile to try to set up a systematic theory for such non-absolutely-integrable integrals that is anywhere near as complete as the absolutely integrable theory, and instead deal with such exotic integrals on an ad hoc basis. From the pointwise triangle inequality |f(x)+g(x)| ≤ |f(x)|+|g(x)|, we conclude the L1 triangle inequality (1.13) f + g L1(Rd) ≤ f L1(Rd) + g L1(Rd) for any almost everywhere defined measurable f, g : Rd → C. It is also easy to see that cf L1(Rd) = |c|f L1(Rd) for any complex number c. As such, we see that L1(Rd → C) is a complex vector space. (The L1 norm is then a seminorm on this space see §1.3 of An epsilon of room, Vol. I.) From Exercise 1.3.18 we make the important observation that a function f ∈ L1(Rd → C) has zero L1 norm, f L1(Rd) = 0, if and only if f is zero almost everywhere. Given two functions f, g ∈ L1(Rd → C), we can define the L1 distance dL1 (f, g) between them by the formula dL1 (f, g) := f − g L1(Rd) . Thanks to (1.13), this distance obeys almost all the axioms of a metric on L1(Rd), with one exception: it is possible for two different functions f, g ∈ L1(Rd → C) to have a zero L1 distance, if they agree almost everywhere. As such, dL1 is only a semi-metric (also known as a pseudo-metric) rather than a metric. However, if one adopts the convention that any two functions that agree almost everywhere are considered equivalent (or more formally,

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