By Tao T.

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Qk ) = m(Q1 ) + . . + m(Qk ) ≥ m(Q1 ) + . . + m(Qk ) − ε = m(E) − ε. 2. Lebesgue measure 27 Applying by monotonicity of Lebesgue outer measure, we conclude that m∗ (E) ≥ m(E) − ε for every ε > 0. Since ε > 0 was arbitrary, the claim follows. The above lemma allows us to compute the Lebesgue outer measure of a finite union of boxes. From this and monotonicity we conclude that the Lebesgue outer measure of any set is bounded below by its Jordan inner measure. 2) for every E ⊂ Rd . 8. We are now able to explain why not every bounded open set or compact set is Jordan measurable.

2. Lebesgue measure 21 and the Lebesgue measure and Jordan measure of a Jordan measurable set are always equal. We will also see a few other equivalent descriptions of the concept of Lebesgue measurability. In the notes below we will establish the basic properties of Lebesgue measure. Broadly speaking, this concept obeys all the intuitive properties one would ask of measure, so long as one restricts attention to countable operations rather than uncountable ones, and as long as one restricts attention to Lebesgue measurable sets.

Note that the upper and lower Darboux integrals are related by the reflection identity b b − f (x) dx = − a f (x) dx. 22. Let [a, b] be an interval, and f : [a, b] → R be a bounded function. Show that f is Riemann integrable if and only if it is Darboux integrable, in which case the Riemann integral and Darboux integrals are equal. 23. Show that any continuous function f : [a, b] → R is Riemann integrable. More generally, show that any bounded, piecewise continuous8 function f : [a, b] → R is Riemann integrable.

### An introduction to measure theory by Tao T.

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