Absolute Continuity of Lebesgue Integral

The following is a wonderful property of the Lebesgue Integral, also known as absolute continuity of Lebesgue Integral. Basically, it means that whenever the domain of integration has small enough measure, then the integral will be arbitrarily small. Suppose is integrable. Given , there exists such that for all measurable sets with , . Proof: … Continue reading “Absolute Continuity of Lebesgue Integral”

Lebesgue’s Dominated Convergence Theorem for Convergence in Measure

Lebesgue’s Dominated Convergence Theorem for Convergence in Measure If satisfies on and , then and . Proof Let be any subsequence of . Then on . Thus there is a subsequence a.e.\ in . Clearly . By the usual Lebesgue’s DCT, and . Since every subsequence of has a further subsequence that converges to , … Continue reading “Lebesgue’s Dominated Convergence Theorem for Convergence in Measure”

Generalized Lebesgue Dominated Convergence Theorem Proof

This key theorem showcases the full power of Lebesgue Integration Theory. Generalized Lebesgue Dominated Convergence Theorem Let and be sequences of measurable functions on satisfying a.e. in , a.e. in , and a.e. in . If and , then . Proof We have . Applying Fatou’s lemma to the non-negative sequence we get That is, Since , … Continue reading “Generalized Lebesgue Dominated Convergence Theorem Proof”

Inner and Outer Approximation of Lebesgue Measurable Sets

Let . Then each of the following four assertions is equivalent to the measurability of . (Outer Approximation by Open Sets and Sets) (i) For each , there is an open set containing for which . (ii) There is a set containing for which . (Inner Approximation by Closed Sets and Sets) (iii) For each … Continue reading “Inner and Outer Approximation of Lebesgue Measurable Sets”

Weierstrass M Test and Lebesgue’s Dominated Convergence Theorem

Previously, we wrote a blog post about Weierstrass M Test. It turns out Weierstrass M Test is a special case of Lebesgue’s Dominated Convergence Theorem, a very powerful theorem in Measure Theory, where the measure is taken to be the counting measure. Lebesgue Dominated Convergence Theorem: Let be a sequence of integrable functions which converges a.e. to a … Continue reading “Weierstrass M Test and Lebesgue’s Dominated Convergence Theorem”

Relationship between L^p convergence and a.e. convergence

It turns out that convergence in Lp implies that the norms converge. Conversely, a.e. convergence and the fact that norms converge implies Lp convergence. Amazing! Relationship between convergence and a.e. convergence: Let , . If , then . Conversely, if a.e.\ and , , then . Note that the converse may fail for . Proof: … Continue reading “Relationship between L^p convergence and a.e. convergence”

Wheeden Zygmund Measure and Integration Solutions

Here are some solutions to exercises in the book: Measure and Integral, An Introduction to Real Analysis by Richard L. Wheeden and Antoni Zygmund. Chapter 1,2: analysis1 Chapter 3: analysis2 Chapter 4, 5: analysis3 Chapter 5,6: analysis4 Chapter 6,7: analysis5 Chapter 8: analysis6 Chapter 9: analysis7 Measure and Integral: An Introduction to Real Analysis, Second Edition (Chapman & Hall/CRC Pure and Applied … Continue reading “Wheeden Zygmund Measure and Integration Solutions”

Leibniz Integral Rule (Differentiating under Integral) + Proof

“Differentiating under the Integral” is a useful trick, and here we describe and prove a sufficient condition where we can use the trick. This is the Measure-Theoretic version, which is more general than the usual version stated in calculus books. Let be an open subset of , and be a measure space. Suppose satisfies the … Continue reading “Leibniz Integral Rule (Differentiating under Integral) + Proof”

The most Striking Theorem in Real Analysis

Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result. Lebesgue’s Theorem (Monotone functions) If the function is monotone on the open interval , then it is differentiable almost everywhere on . Absolutely Continuous Functions Definition A real-valued function on a closed, … Continue reading “The most Striking Theorem in Real Analysis”

Outer Measure Zero implies Measurable

One quick way is to use Caratheodory’s Criterion: Let denote the Lebesgue outer measure on , and let . Then is Lebesgue measurable if and only if for every . Suppose is a set with outer measure zero, and be any subset of . Then by the monotonicity of outer measure. The other direction follows … Continue reading “Outer Measure Zero implies Measurable”

Outer measure of Symmetric Difference Zero implies Measurability

Free Career Quiz: Please help to do! Just came across this neat beginner’s Lebesgue Theory question. As students of analysis know, just to show a set is measurable is no easy feat. The usual way is to use the Caratheodory definition, where  a set E is said to be measurable if for any set A, … Continue reading “Outer measure of Symmetric Difference Zero implies Measurability”

Measure Theory: What does a.e. (almost everywhere) mean

Source: Elements of Integration by Professor Bartle Students studying Mathematical Analysis, Advanced Calculus, or probability would sooner or later come across the term a.e. or “almost everywhere”. In layman’s terms, it means that the proposition (in the given context) holds for all cases except for a certain subset which is very small. For instance, if … Continue reading “Measure Theory: What does a.e. (almost everywhere) mean”

Lemma on Measure of Increasing Sequences in X

(Continued from https://mathtuition88.com/2015/06/20/what-is-a-measure-measure-theory/) Lemma: Let be a measure defined on a -algebra X. (a) If () is an increasing sequence in X, then (b) If ) is a decreasing sequence in X and if , then Note: An increasing sequence of sets () means that for all natural numbers n, . A decreasing sequence means … Continue reading “Lemma on Measure of Increasing Sequences in X”

What is a Measure? (Measure Theory)

In layman’s terms, “measures” are functions that are intended to represent ideas of length, area, mass, etc. The inputs for the measure functions would be sets, and the output would be a real value, possibly including infinity. It would be desirable to attach the value 0 to the empty set and measures should be additive … Continue reading “What is a Measure? (Measure Theory)”

Functions between Measurable Spaces

Sometimes, it is  desirable to define measurability for a function f from one measurable space (X,X) into another measurable space (Y,Y). In this case one can define f to be measurable if and only if the set belongs to X for every set E belonging to Y. This definition of measurability appears to differ from … Continue reading “Functions between Measurable Spaces”

A nonnegative function f in M(X,X) is the limit of a monotone increasing sequence in M(X,X)

We will elaborate on a lemma in the book The Elements of Integration and Lebesgue Measure. Lemma: If f is a nonnegative function in M(X,X), then there exists a sequence () in M(X,X) such that: (a) for . (b) for each . (c) Each has only a finite number of real values. Proof: Let n … Continue reading “A nonnegative function f in M(X,X) is the limit of a monotone increasing sequence in M(X,X)”

Measurability of product fg

In the previous chapters, Bartle showed that that if f is in M(X,X), then the functions are also in M(X,X). The case of the measurability of the product fg when f, g belong to M(X,X) is a little bit more tricky. If , let be the “truncation of f” defined by Let be defined similarly. … Continue reading “Measurability of product fg”

Definitions for measurable functions

I am currently proceeding on a self-guided study of the book The Elements of Integration and Lebesgue Measure, will post some updates and elaborations of the proofs in the book. Every book is constrained by the number of pages the publisher allows, hence some authors will write rather terse and concise proofs, the worst example … Continue reading “Definitions for measurable functions”

Measure and Integration Recommended Book

I have added a new addition to the Recommended Books for Undergraduate Math, which is one of my most popular posts! The new book is The Elements of Integration and Lebesgue Measure, an advanced text on the theory of integration. At the high school level, students are exposed to integration, but merely the rules of … Continue reading “Measure and Integration Recommended Book”

Undergraduate Level Math Book Recommendations

Highly Recommended Math Books for University Self Study Recently, a viewer of my website asked if I was able to suggest any undergraduate level university textbooks for self study that follows the university curriculum. Self-study is challenging but not impossible. Choosing a good and appropriate book of the right level is of crucial importance. For … Continue reading “Undergraduate Level Math Book Recommendations”