By Terence Tao

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**Extra info for An Introduction To Measure Theory (January 2011 Draft)**

**Example text**

The above definition, while geometrically natural, can be awkward to use in practice. A more convenient formulation of the Riemann integral can be formulated using some additional machinery. 1. 20 (Piecewise constant functions). Let [a, b] be an interval. A piecewise constant function f : [a, b] → R is a function for which there exists a partition of [a, b] into finitely many intervals I1 , . . , In , such that f is equal to a constant ci on each of the intervals Ii . If f is piecewise constant, show that the expression n ci |Ii | i=1 is independent of the choice of partition used to demonstrate the piecewise constant nature of f .

44). Once we have the theory of the unsigned Lebesgue integral, we will then be able to define the absolutely convergent Lebesgue integral, similarly to how the absolutely convergent infinite sum can be defined using the unsigned infinite sum. 49). This convergence theorem makes the Lebesgue integral (and its abstract generalisations to other measure spaces than Rd ) particularly suitable for analysis, as well as allied fields that rely heavily on limits of functions, such as PDE, probability, and ergodic theory.

N=1 Letting N → ∞ we obtain the claim. 10. 2. Lebesgue measure ∞ n=1 29 ∞ |Bn | = n=1 |Bn |. Although this statement is intuitively obvious and does not explicitly use the concepts of Lebesgue outer measure or Lebesgue measure, it is remarkably difficult to prove this statement rigorously without essentially using one of these two concepts. 5. Show that if a set E ⊂ Rd is expressible as the countable union of almost disjoint boxes, then the Lebesgue outer measure of E is equal to the Jordan inner measure: m∗ (E) = m∗,(J) (E), where we extend the definition of Jordan inner measure to unbounded sets in the obvious manner.