Function Theory on Planar Domains: A Second Course in by Stephen D. Fisher

By Stephen D. Fisher

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Extra resources for Function Theory on Planar Domains: A Second Course in Complex Analysis (Pure & Applied Mathematics)

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Thus m∗ (L(E)) ≤ c , and hence m(L(E)) = 0. 5. 40 Chapter 1. MEASURE THEORY One can show that m(L(E)) = | det L| m(E); see Problem 4 in the next chapter. 9. Give an example of an open set O with the following property: the boundary of the closure of O has positive Lebesgue measure. ] 10. This exercise provides a construction of a decreasing sequence of positive continuous functions on the interval [0, 1], whose pointwise limit is not Riemann integrable. Let Cˆ denote a Cantor-like set obtained from the construction detailed in Exerˆ > 0.

A calculation of the constant vd is postponed until Exercise 14 in the next chapter. 7. If δ = (δ1 , . . , δd ) is a d-tuple of positive numbers δi > 0, and E is a subset of Rd , we define δE by δE = {(δ1 x1 , . . , δd xd ) : where (x1 , . . , xd ) ∈ E}. Prove that δE is measurable whenever E is measurable, and m(δE) = δ1 · · · δd m(E). 8. Suppose L is a linear transformation of Rd . Show that if E is a measurable subset of Rd , then so is L(E), by proceeding as follows: (a) Note that if E is compact, so is L(E).

For instance, C is closed and bounded, hence compact. Also, C is totally disconnected: given any x, y ∈ C there exists z ∈ / C that lies between x and y. Finally, C is perfect: it has no isolated points (Exercise 1). Next, we turn our attention to the question of determining the “size” of C. This is a delicate problem, one that may be approached from different angles depending on the notion of size we adopt. For instance, in terms of cardinality the Cantor set is rather large: it is not countable.

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