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Unformatted text preview: CONTINUOUS ALMOST EVERYWHERE Definition 1. Let Δ be a subset of R . We say that Δ has measure if, for each ǫ > , there is a sequence of balls ( B j = B r j ( c j ))) j ∈ N with radii r j > (and centres c j ∈ R ), such that Δ ⊂ uniontext j ∈ N B j and ∑ ∞ j =1 r j < ǫ . The balls here are open intervals of lenght 2 r j , so the series ∑ ∞ j =1 2 r j < 2 ǫ is the sum of lengths. However, the balls could intersect each other, so 2 ǫ is just an upper bound for the ”total length” or ”volume” (what we will call ”measure”) of their union uniontext j B j . A full definition of ”measures” is beyond the scope of this course; the only important property that we will use here is that subsets have smaller measure than the set that they are contained in (assuming both are ”measurable”). So the above Definition just says that the measure of Δ is smaller than 2 ǫ for any ǫ > . So – if the measure of Δ is to be a nonnegative real number – the measure of Δ will indeed be . Examples of measure subsets of R are all finite and countable subsets (see Lemma below) as well as the (uncountable) Cantor set. Lemma 2. Let Δ ⊂ R be a countable subset, then Δ has measure . Proof. By assumption, we can enumerate Δ = { c j  j ∈ N } . Now for any ǫ > choose the sequence of radii r j = 2 − j − 1 ǫ , then clearly uniontext j ∈ N B r j ( c j ) contains Δ (since it contains every c j ∈ Δ as centre of a ball) and we have ∑ ∞ j =1 r j = ∑ ∞ k =0 2 − k ǫ 4 = ǫ 2 < ǫ . Given a function f : [ a, b ] → R , we say it is continuous almost everywhere if the set Δ f ⊂ [ a, b ] of discontinuities of f has measure . If the set of discontinuities is finite or countable, then f is continuous almost everywhere, for example. So the function f ( x ) = 1 /x for x negationslash = 0 and for x = 0 is continuous almost everywhere. However, this function is not Riemannis continuous almost everywhere....
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
 Fall '10
 Prof.KatrinWehrheim

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