60 1. Measure theory for some real θ. Taking real parts of both sides, we obtain | Rd f(x) dx| = Rd Re(eiθf(x)) dx. Since Re(eiθf(x)) |eiθf(x)| = |f(x)|, we obtain the claim. 1.3.5. Littlewood’s three principles. Littlewood’s three principles are informal heuristics that convey much of the basic intuition behind the mea- sure theory of Lebesgue. Briefly, the three principles are as follows: (i) Every (measurable) set is nearly a finite sum of intervals (ii) Every (absolutely integrable) function is nearly continuous and (iii) Every (pointwise) convergent sequence of functions is nearly uni- formly convergent. Various manifestations of the first principle were given in Exercise 1.2.7 and Exercise 1.2.16. Now we turn to the second principle. Define a step function to be a finite linear combination of indicator functions 1B of boxes B. Theorem 1.3.20 (Approximation of L1 functions). Let f L1(Rd) and ε 0. (i) There exists an absolutely integrable simple function g such that f g L1(Rd) ε. (ii) There exists a step function g such that f g L1(Rd) ε. (iii) There exists a continuous, compactly supported g such that f g L1(Rd) ε. To put things another way, the absolutely integrable simple functions, the step functions, and the continuous, compactly supported functions are all dense subsets of L1(Rd) with respect to the L1(Rd) (semi-)metric. In §1.13 of An epsilon of room, Vol. I, it is shown that a similar statement holds if one replaces continuous, compactly supported functions with smooth, compactly supported functions, also known as test functions this is an important fact for the theory of distributions. Proof. We begin with part (i). When f is unsigned, we see from the def- inition of the lower Lebesgue integral that there exists an unsigned simple function g such that g f (so, in particular, g is absolutely integrable) and Rd g(x) dx Rd f(x) dx ε, which by linearity implies that f g L1(Rd) ε. This gives (i) when f is unsigned. The case when f is real-valued then follows by splitting f
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