By Debarra G.
Techniques integration through degree, instead of degree through integration.
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- E Measure Theory
- Seminaire Pierre Lelong (Analyse) Annee 1970-1971
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Extra resources for Measure Theory and Integration
Sample text
Our results on approximation to measurable sets by intervals or by open sets lead to results on approximation to the integral of a measurable function. In the last section we compare the Lebesgue and Riemann integrals. 3 . 1 INTEGRATION OF NON-NEGATIVE FUNCTIONS We consider first the class of non-negative measurable functions, defme the integral of such a function and examine the properties of the integral. For the present we will suppose these functions to be defined for all real x . A non-negative fmite-valutd function �x), taking only a finite number of an are the distinct different values, is called a simple function.
Since 0 1 is countable, each Eo: is a countable set. Since [0, 1 ] is uncountable there are uncountable many sets Ea . Using the Axiom of Choice, p. 1 7 , we consider a set V in [0, 1 ] containing just one element Xo: from each Ea . Let {r;} be an enumeration of 0 1 , and for each n write Vn = V + rn . Ify E Vn n V there exist X a , x13 E V such that y = Xa + rn and y = x13 + r But then x13 X a E 0 1 , so x13 = Xo: by defmition of V and we have n = m . So Vn n V = rv m m . m (/J sec. 2 . 5 J Borel and Lebesgue Measurability 43 00 for n =l= m .
D Exercises 46 . Show that every countable set has Hausdorff dimension zero. 47. Let h(t) = sin t (0 < t < rr / 2) . Show that, for any Borel set A , H(A ) = m(A). 4 8. Let {An } be Borel sets and let an be the Hausdorff dimension of A n . Find 00 the Hausdorff dimension of A = U An . n= l 49. Show that , for 0 < 8 < 1 , n;,6 (A ) is a mono tone decreasing fun cti o n of p . 50. Show that, for 0 < q < 1 , we may construct a set A as in Example 26 , but which is compact ) so that Hp(A) = oo (0 < p � q), Hp(A ) = 0 (p > q).
