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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.

A set E ⊂ Rd is said to be Lebesgue measurable if, for every ε > 0, there exists an open set U ⊂ Rd containing E such that m∗ (U \E) ≤ ε. If E is Lebesgue measurable, we refer to m(E) := m∗ (E) as the Lebesgue measure of E (note that this quantity may be equal to +∞). We also write m(E) as md (E) when we wish to emphasise the dimension d. 3. The intuition that measurable sets are almost open is also known as Littlewood’s first principle, this principle is a triviality with our current choice of definitions, though less so if one uses other, equivalent, definitions of Lebesgue measurability.

Show that m : L/ ∼→ R+ is the unique continuous extension of the analogous elementary measure function m : E/ ∼→ R+ to L/ ∼. 1 of An epsilon of room, Vol. I. 25. Define a continuously differentiable curve in Rd to be a set of the form {γ(t) : a ≤ t ≤ b} where [a, b] is a closed interval and γ : [a, b] → Rd is a continuously differentiable function. (i) If d ≥ 2, show that every continuously differentiable curve has Lebesgue measure zero. ) (ii) Conclude that if d ≥ 2, then the unit cube [0, 1]d cannot be covered by countably many continuously differentiable curves.

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