Page 284 - DMTH401_REAL ANALYSIS

Real Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 23: Differentiation of Integrals

CONTENTS
Objectives
Introduction
23.1 Differentiation of Integrals
23.2 Theorems on the Differentiation of Integrals
23.3 Summary

23.4 Keywords
23.5 Review Questions
23.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Define Differentiation of Integrals
 Discuss the Theorems on the Differentiation of Integrals
Introduction

In this unit, we are going to study about differentiation of integrals. Suppose  is a function of
two variables which can be integrated with respect to one variable and which can be differentiated
with respect to another variable. We are going to see under what conditions the result will be
the same if these two limit process are carried out in the opposite order.

23.1 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. More formally, given a space X with
a measure  and a metric d, one asks for what functions f : X  R does

1

lim ò f(y)d (y) = f(x)
r u (B (x)) r B (x)
r
for all (or at least -almost all) x  X? (Here, as in the rest of the article, B (x) denotes the open ball
r
in X with d-radius r and centre x.) This is a natural question to ask, especially in view of the
heuristic construction of the Riemann integral, in which it is almost implicit that f(x) is a “good
representative” for the values of f near x.
23.2 Theorems on the Differentiation of Integrals

Lebesgue Measure

One result on the differentiation of integrals is the Lebesgue differentiation theorem, as proved
n
by Henri Lebesgue in 1910. Consider n-dimensional Lebesgue measure  on n-dimensional
n
n
Euclidean space R . Then, for any locally integrable function f : R  R, one has

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