38 1. Measure theory On the other hand, the sum q∈Q∩[−1,1] m(E) is either zero (if m(E) = 0) or infinite (if m(E) 0), leading to the desired contradiction. Exercise 1.2.26 (Outer measure is not finitely additive). Show that there exists disjoint bounded subsets E, F of the real line such that m∗(E F ) = m∗(E) + m∗(F ). (Hint: Show that the set constructed in the proof of the above proposition has positive outer measure.) Exercise 1.2.27 (Projections of measurable sets need not be measurable). Let π : R2 R be the coordinate projection π(x, y) := x. Show that there exists a measurable subset E of R2 such that π(E) is not measurable. Remark 1.2.19. The above discussion shows that, in the presence of the axiom of choice, one cannot hope to extend Lebesgue measure to arbitrary subsets of R while retaining both the countable additivity and the trans- lation invariance properties. If one drops the translation invariant require- ment, then this question concerns the theory of measurable cardinals, and is not decidable from the standard ZFC axioms. On the other hand, one can construct finitely additive translation invariant extensions of Lebesgue measure to the power set of R by use of the Hahn-Banach theorem (§1.5 of An epsilon of room, Vol. I ) to extend the integration functional, though we will not do so here. 1.3. The Lebesgue integral In Section 1.2, we defined the Lebesgue measure m(E) of a Lebesgue mea- surable set E Rd, and set out the basic properties of this measure. In this set of notes, we use Lebesgue measure to define the Lebesgue integral Rd f(x) dx of functions f : Rd C {∞}. Just as not every set can be measured by Lebesgue measure, not every function can be integrated by the Lebesgue integral the function will need to be Lebesgue measurable. Furthermore, the function will either need to be unsigned (taking values on [0, +∞]), or absolutely integrable. To motivate the Lebesgue integral, let us first briefly review two simpler integration concepts. The first is that of an infinite summation n=1 cn of a sequence of numbers cn, which can be viewed as a discrete analogue of the Lebesgue integral. Actually, there are two overlapping, but different, notions of summation that we wish to recall here. The first is that of the
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