Page 292 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 23.4 Keywords

Differentiation of Integrals: In mathematics, the problem of differentiation of integrals is that of
determining under what circumstances the mean value integral of a suitable function on a small
neighbourhood of a point approximates the value of the function at that point.
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Borel measures on R : The result for Lebesgue measure turns out to be a special case of the
following result, which is based on the Besicovitch covering theorem: if  is any locally finite
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Borel measure on R and f : R  R is locally integrable with respect to , then
1
lim ò f(y)d (y) = f(x)

r 0 (B (x)) r B (x)
r
for -almost all points x  R .
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23.5 Review Questions

1. Explain Differentiation of Integrals with the help of example.

2. Discuss the Theorems on the differentiation of integrals.

Answers: Self Assessment

1. 1910 2. Lebesgue measure
3. differentiation of integrals 4. Gaussian measure 

23.6 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).
G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.
Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)

T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.
S.C. Malik: Mathematical Analysis.
H.L. Royden: Real Analysis, Ch. 3, 4.

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