1.3. The Lebesgue integral 39 unsigned infinite sum, when the cn lie in the extended non-negative real axis [0, +∞]. In this case, the infinite sum can be defined as the limit of the partial sums (1.6) ∞ n=1 cn = lim N→∞ N n=1 cn or equivalently as a supremum of arbitrary finite partial sums: (1.7) ∞ n=1 cn = sup A⊂N,A finite n∈A cn. The unsigned infinite sum ∑∞ n=1 cn always exists, but its value may be infi- nite, even when each term is individually finite (consider e.g. ∑∞ n=1 1). The second notion of a summation is the absolutely summable infinite sum, in which the cn lie in the complex plane C and obey the absolute summability condition ∞ n=1 |cn| ∞, where the left-hand side is of course an unsigned infinite sum. When this occurs, one can show that the partial sums ∑N n=1 cn converge to a limit, and we can then define the infinite sum by the same formula (1.6) as in the unsigned case, though now the sum takes values in C rather than [0, +∞]. The absolute summability condition confers a number of useful properties that are not obeyed by sums that are merely conditionally convergent most notably, the value of an absolutely convergent sum is unchanged if one re- arranges the terms in the series in an arbitrary fashion. Note also that the absolutely summable infinite sums can be defined in terms of the unsigned infinite sums by taking advantage of the formulae ∞ n=1 cn = ( ∞ n=1 Re(cn)) + i( ∞ n=1 Im(cn)) for complex absolutely summable cn, and ∞ n=1 cn = ∞ n=1 cn + − ∞ n=1 cn− for real absolutely summable cn, where cn + := max(cn, 0) and cn − := max(−cn, 0) are the (magnitudes of the) positive and negative parts of cn. In an analogous spirit, we will first define an unsigned Lebesgue integral Rd f(x) dx of (measurable) unsigned functions f : Rd → [0, +∞], and then use that to define the absolutely convergent Lebesgue integral Rd f(x) dx of absolutely integrable functions f : Rd → C∪{∞}. (In contrast to absolutely summable series, which cannot have any infinite terms, absolutely integrable

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