Unformatted text preview: Modern Analysis 2
Test 4
Answer TWO questions.
1. Let f ∈ L(µ) and deﬁne φ : M → R by
A ∈ M ⇒ φ(A) = f dµ.
A Show that if ε > 0 is given then there exists δ > 0 such that
µ(A) < δ ⇒ φ(A) < ε.
2. Let f : R → R be Lebesgue integrable. For t ∈ R deﬁne
t f F (t) =
0 with respect to Lebesgue measure. Prove that F is continuous.
3. Show that if a ∈ R then
∞ 2 /2 e−t cos(at)dt = 0 1 2 /2 π /2 e−a . ...
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 Spring '11
 Robinson
 µ, measure, Lebesgue measure, Lebesgue integration, Modern Analysis

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