1.3. The Lebesgue integral 61 into positive and negative parts (and adjusting ε as necessary), and the case when f is complex-valued then follows by splitting f into real and imaginary parts (and adjusting ε yet again). To establish part (ii), we see from (i) and the triangle inequality in L1 that it suffices to show this when f is an absolutely integrable simple function. By linearity (and more applications of the triangle inequality), it then suffices to show this when f = 1E is the indicator function of a measurable set E Rd of finite measure. But then, by Exercise (1.2.16), such a set can be approximated (up to an error of measure at most ε) by an elementary set, and the claim follows. To establish part (iii), we see from (ii) and the argument from the pre- ceding paragraph that it suffices to show this when f = 1E is the indicator function of a box. But one can then establish the claim by direct construc- tion. Indeed, if one makes a slightly larger box F that contains the closure of E in its interior, but has a volume at most ε more than that of E, then one can directly construct a piecewise linear continuous function g supported on F that equals 1 on E (e.g. one can set g(x) = max(1 R dist(x, E), 0) for some sufficiently large R one may also invoke Urysohn’s lemma, see §1.10 of An epsilon of room, Vol. I ). It is then clear from construction that f g L1(Rd) ε as required. This is not the only way to make Littlewood’s second principle manifest we return to this point shortly. For now, we turn to Littlewood’s third principle. We recall three basic ways in which a sequence fn : Rd C of functions can converge to a limit f : Rd C: (i) (Pointwise convergence) fn(x) f(x) for every x Rd. (ii) (Pointwise almost everywhere convergence) fn(x) f(x) for al- most every x Rd. (iii) (Uniform convergence) For every ε 0, there exists N such that |fn(x) f(x)| ε for all n N and all x Rd. Uniform convergence implies pointwise convergence, which in turn im- plies pointwise almost everywhere convergence. We now add a fourth mode of convergence, that is weaker than uniform convergence but stronger than pointwise convergence: Definition 1.3.21 (Locally uniform convergence). A sequence of functions fn : Rd C converges locally uniformly to a limit f : Rd C if, for every bounded subset E of Rd, fn converges uniformly to f on E. In other words, for every bounded E Rd and every ε 0, there exists N 0 such that |fn(x) f(x)| ε for all n N and x E.
Previous Page Next Page