Topics in Measure Theory and Real Analysis: The Measure Extension Problem and Related Questions

Unfortunately, none of the other modes of convergence automatically imply this convergence of the integral, as the above examples show. The purpose of these notes is to compare these modes of convergence with each other. Unfortunately, the relationship between these modes is not particularly simple; unlike the situation with pointwise and uniform convergence, one cannot simply rank these modes in a linear order from strongest to weakest.

On the other hand, if one imposes some additional assumptions to shut down one or more of these escape to infinity scenarios, such as a finite measure hypothesis or a uniform integrability hypothesis , then one can obtain some additional implications between the different modes. CA Tags: absolutely integrable functions , Fatou's lemma , Lebesgue dominated convergence theorem , measurable space , measure space , monotone convergence theorem , sigma-algebra by Terence Tao 99 comments.

Thus far, we have only focused on measure and integration theory in the context of Euclidean spaces. Now, we will work in a more abstract and general setting, in which the Euclidean space is replaced by a more general space. It turns out that in order to properly define measure and integration on a general space , it is not enough to just specify the set.

One also needs to specify two additional pieces of data:. For instance, Lebesgue measure theory covers the case when is a Euclidean space , is the collection of all Lebesgue measurable subsets of , and is the Lebesgue measure of. The collection has to obey a number of axioms e.

Similarly, the measure has to obey a number of axioms most notably, a countable additivity axiom in order to obtain a measure and integration theory comparable to the Lebesgue theory on Euclidean spaces. When all these axioms are satisfied, the triple is known as a measure space.

These play much the same role in abstract measure theory that metric spaces or topological spaces play in abstract point-set topology , or that vector spaces play in abstract linear algebra. One question that will not be addressed much in this current set of notes is how one actually constructs interesting examples of measures.

We will discuss this issue more in later notes although one of the most powerful tools for such constructions, namely the Riesz representation theorem , will not be covered until B. CA Tags: absolute integrability , Egorov's theorem , Lebesgue integration , Littlewood's three principles , Lusin's theorem by Terence Tao comments. In the previous notes, we defined the Lebesgue measure of a Lebesgue measurable set , and set out the basic properties of this measure. In this set of notes, we use Lebesgue measure to define the Lebesgue integral.

Lebesgue outer measure: definition and properties (MAT)

Just as not every set can be measured by Lebesgue measure, not every function can be integrated by the Lebesgue integral; the function will need to be Lebesgue measurable. Furthermore, the function will either need to be unsigned taking values on , or absolutely integrable.

To motivate the Lebesgue integral, let us first briefly review two simpler integration concepts. The first is that of an infinite summation. Actually, there are two overlapping, but different, notions of summation that we wish to recall here. The first is that of the unsigned infinite sum , when the lie in the extended non-negative real axis. In this case, the infinite sum can be defined as the limit of the partial sums. The unsigned infinite sum always exists, but its value may be infinite, even when each term is individually finite consider e.

The second notion of a summation is the absolutely summable infinite sum , in which the lie in the complex plane and obey the absolute summability condition. When this occurs, one can show that the partial sums converge to a limit, and we can then define the infinite sum by the same formula 1 as in the unsigned case, though now the sum takes values in rather than.

The absolute summability condition confers a number of useful properties that are not obeyed by sums that are merely conditionally convergent; most notably, the value of an absolutely convergent sum is unchanged if one rearranges the terms in the series in an arbitrary fashion. Note also that the absolutely summable infinite sums can be defined in terms of the unsigned infinite sums by taking advantage of the formulae. In an analogous spirit, we will first define an unsigned Lebesgue integral of measurable unsigned functions , and then use that to define the absolutely convergent Lebesgue integral of absolutely integrable functions.

In contrast to absolutely summable series, which cannot have any infinite terms, absolutely integrable functions will be allowed to occasionally become infinite. However, as we will see, this can only happen on a set of Lebesgue measure zero. To define the unsigned Lebesgue integral, we now turn to another more basic notion of integration, namely the Riemann integral of a Riemann integrable function. Recall from the prologue that this integral is equal to the lower Darboux integral. Compare this formula also with 2. The integral is a piecewise constant integral, formed by breaking up the piecewise constant functions into finite linear combinations of indicator functions of intervals, and then measuring the length of each interval.

It turns out that virtually the same definition allows us to define a lower Lebesgue integral of any unsigned function , simply by replacing intervals with the more general class of Lebesgue measurable sets and thus replacing piecewise constant functions with the more general class of simple functions. If the function is Lebesgue measurable a concept that we will define presently , then we refer to the lower Lebesgue integral simply as the Lebesgue integral. Once we have the theory of the unsigned Lebesgue integral, we will then be able to define the absolutely convergent Lebesgue integral, similarly to how the absolutely convergent infinite sum can be defined using the unsigned infinite sum.

This integral also obeys all the basic properties one expects, such as linearity and compatibility with the more classical Riemann integral; in subsequent notes we will see that it also obeys a fundamentally important convergence theorem, the dominated convergence theorem. This convergence theorem makes the Lebesgue integral and its abstract generalisations to other measure spaces than particularly suitable for analysis, as well as allied fields that rely heavily on limits of functions, such as PDE, probability, and ergodic theory.

Remark 1 This is not the only route to setting up the unsigned and absolutely convergent Lebesgue integrals. Stein-Shakarchi, for instance, proceeds slightly differently, beginning with the unsigned integral but then making an auxiliary stop at integration of functions that are bounded and are supported on a set of finite measure, before going to the absolutely convergent Lebesgue integral. Another approach which will not be discussed here is to take the metric completion of the Riemann integral with respect to the metric.

The Lebesgue integral and Lebesgue measure can be viewed as completions of the Riemann integral and Jordan measure respectively. This means three things. Firstly, the Lebesgue theory extends the Riemann theory: every Jordan measurable set is Lebesgue measurable, and every Riemann integrable function is Lebesgue measurable, with the measures and integrals from the two theories being compatible.

Conversely, the Lebesgue theory can be approximated by the Riemann theory; as we saw in the previous notes , every Lebesgue measurable set can be approximated in various senses by simpler sets, such as open sets or elementary sets, and in a similar fashion, Lebesgue measurable functions can be approximated by nicer functions, such as Riemann integrable or continuous functions. Finally, the Lebesgue theory is complete in various ways; we will formalise this properly only in the next quarter when we study spaces , but the convergence theorems mentioned above already hint at this completeness.

In the prologue for this course, we recalled the classical theory of Jordan measure on Euclidean spaces. This theory proceeded in the following stages:. As long as one is lucky enough to only have to deal with Jordan measurable sets, the theory of Jordan measure works well enough.

However, as noted previously, not all sets are Jordan measurable, even if one restricts attention to bounded sets. In fact, we shall see later in these notes that there even exist bounded open sets, or compact sets, which are not Jordan measurable, so the Jordan theory does not cover many classes of sets of interest. Another class that it fails to cover is countable unions or intersections of sets that are already known to be measurable:.

Exercise 1 Show that the countable union or countable intersection of Jordan measurable sets need not be Jordan measurable, even when bounded. This creates problems with Riemann integrability which, as we saw in the preceding notes, was closely related to Jordan measure and pointwise limits:. Exercise 2 Give an example of a sequence of uniformly bounded, Riemann integrable functions for that converge pointwise to a bounded function that is not Riemann integrable. What happens if we replace pointwise convergence with uniform convergence?

These issues can be rectified by using a more powerful notion of measure than Jordan measure, namely Lebesgue measure. To define this measure, we first tinker with the notion of the Jordan outer measure. Observe from the finite additivity and subadditivity of elementary measure that we can also write the Jordan outer measure as. The natural number is allowed to vary freely in the above infimum. We now modify this by replacing the finite union of boxes by a countable union of boxes, leading to the Lebesgue outer measure of :. Note that the countable sum may be infinite, and so the Lebesgue outer measure could well equal.

Caution: the Lebesgue outer measure is sometimes denoted ; this is for instance the case in Stein-Shakarchi.

Clearly, we always have since we can always pad out a finite union of boxes into an infinite union by adding an infinite number of empty boxes. But can be a lot smaller:. Example 1 Let be a countable set. We know that the Jordan outer measure of can be quite large; for instance, in one dimension, is infinite, and since has as its closure see Exercise 18 of the prologue. On the other hand, all countable sets have Lebesgue outer measure zero. Indeed, one simply covers by the degenerate boxes of sidelength and volume zero.

Alternatively, if one does not like degenerate boxes, one can cover each by a cube of sidelength say for some arbitrary , leading to a total cost of , which converges to for some absolute constant. As can be arbitrarily small, we see that the Lebesgue outer measure must be zero. We will refer to this type of trick as the trick ; it will be used many further times in this course. From this example we see in particular that a set may be unbounded while still having Lebesgue outer measure zero, in contrast to Jordan outer measure.

As we shall see later in this course, Lebesgue outer measure also known as Lebesgue exterior measure is a special case of a more general concept known as an outer measure. Here, there is an asymmetry which ultimately arises from the fact that elementary measure is subadditive rather than superadditive : one does not gain any increase in power in the Jordan inner measure by replacing finite unions of boxes with countable ones.

But one can get a sort of Lebesgue inner measure by taking complements; see Exercise However, this is not the most intuitive formulation of this concept to work with, and we will instead use a different but logically equivalent definition of Lebesgue measurability. The starting point is the observation see Exercise 5 of the prologue that Jordan measurable sets can be efficiently contained in elementary sets, with an error that has small Jordan outer measure.

In a similar vein, we will define Lebesgue measurable sets to be sets that can be efficiently contained in open sets, with an error that has small Lebesgue outer measure:. Definition 1 Lebesgue measurability A set is said to be Lebesgue measurable if, for every , there exists an open set containing such that. If is Lebesgue measurable, we refer to as the Lebesgue measure of note that this quantity may be equal to. We also write as when we wish to emphasise the dimension.

As we shall see later, Lebesgue measure extends Jordan measure, in the sense that every Jordan measurable set is Lebesgue measurable, and the Lebesgue measure and Jordan measure of a Jordan measurable set are always equal. We will also see a few other equivalent descriptions of the concept of Lebesgue measurability.

In the notes below we will establish the basic properties of Lebesgue measure. Broadly speaking, this concept obeys all the intuitive properties one would ask of measure, so long as one restricts attention to countable operations rather than uncountable ones, and as long as one restricts attention to Lebesgue measurable sets. The latter is not a serious restriction in practice, as almost every set one actually encounters in analysis will be measurable the main exceptions being some pathological sets that are constructed using the axiom of choice.

In the next set of notes we will use Lebesgue measure to set up the Lebesgue integral, which extends the Riemann integral in the same way that Lebesgue measure extends Jordan measure; and the many pleasant properties of Lebesgue measure will be reflected in analogous pleasant properties of the Lebesgue integral most notably the convergence theorems.

We will treat all dimensions equally here, but for the purposes of drawing pictures, we recommend to the reader that one sets equal to. However, for this topic at least, no additional mathematical difficulties will be encountered in the higher-dimensional case though of course there are significant visual difficulties once exceeds. The material here is based on Sections 1. CA Tags: elementary sets , Jordan measure , piecewise constant functions , Riemann integral by Terence Tao 76 comments.

One of the most fundamental concepts in Euclidean geometry is that of the measure of a solid body in one or more dimensions. In one, two, and three dimensions, we refer to this measure as the length , area , or volume of respectively. In the classical approach to geometry, the measure of a body was often computed by partitioning that body into finitely many components, moving around each component by a rigid motion e.

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One could also obtain lower and upper bounds on the measure of a body by computing the measure of some inscribed or circumscribed body; this ancient idea goes all the way back to the work of Archimedes at least. Such arguments can be justified by an appeal to geometric intuition, or simply by postulating the existence of a measure that can be assigned to all solid bodies , and which obeys a collection of geometrically reasonable axioms.

With the advent of analytic geometry , however, Euclidean geometry became reinterpreted as the study of Cartesian products of the real line. One can also pose the problem of measure on other domains than Euclidean space, such as a Riemannian manifold, but we will focus on the Euclidean case here for simplicity.

To see why this problem exists at all, let us try to formalise some of the intuition for measure discussed earlier. To make matters worse, two bodies that have exactly the same number of points, need not have the same measure. For instance, in one dimension, the intervals and are in one-to-one correspondence using the bijection from to , but of course is twice as long as. So one can disassemble into an uncountable number of points and reassemble them to form a set of twice the length. Of course, one can point to the infinite and uncountable number of components in this disassembly as being the cause of this breakdown of intuition, and restrict attention to just finite partitions.

But one still runs into trouble here for a number of reasons, the most striking of which is the Banach-Tarski paradox , which shows that the unit ball in three dimensions can be disassembled into a finite number of pieces in fact, just five pieces suffice , which can then be reassembled after translating and rotating each of the pieces to form two disjoint copies of the ball. The paradox only works in three dimensions and higher, for reasons having to do with the property of amenability ; see this blog post for further discussion of this interesting topic, which is unfortunately too much of a digression from the current subject.

Here, the problem is that the pieces used in this decomposition are highly pathological in nature; among other things, their construction requires use of the axiom of choice. This is in fact necessary; there are models of set theory without the axiom of choice in which the Banach-Tarski paradox does not occur, thanks to a famous theorem of Solovay. Such pathological sets almost never come up in practical applications of mathematics.

The problem of measure then divides into several subproblems:. These questions are somewhat open-ended in formulation, and there is no unique answer to them; in particular, one can expand the class of measurable sets at the expense of losing one or more nice properties of measure in the process e.

However, it is indeed true Lang's Algebra is a fantastic book, but for students with weak background it is hard to read. This is why there are other seperated book recommendations. Please note that all the books recommended below are for you as a reference when you are lack of background or lose track while reading Lang.

Therefore, Lang should still be the primariry source for you to learn. Lane and G. Atiyah and I. Zariski and P. Gadea, J. Masque and I. Here are few words of mine which I think will be good for a student who is preparing or will prepare the GRE Sub:. Prepare Early. The best time of your first prepartion should be the summer after your Sophormore, and you need to begin taking GRE Math Sub through your Junior year and Senior year So, 6 times. Yes, you will argue that at that time, you will know nothing. But that is okay. This will give you 3 more times to try, and for exams, you need to try as many as possible.

First of all, if you want to get beyond , you need to know basicaly everything. Secondly, even the calculus is not dumb if I ask you to give me the formula of arc length of the function after parametrization and you cannot give me that in 5 seconds, calculus is not really for you. Thirdly, if you take the test, you will find a fact that you cannot get calculus, differential equation and linear algebra all right, and you cannot get points from other things.

In this way, the test is really hard. So, do not overlook this test, it is a hard test. Prepare everything, and in a diffcult level. The difficulty is even getting harder in one year gradually increasing during the three tests. Thus, you need to basically prepare everything, and you need to look for diffcult questions to solve. In this way, you can expand your knowledge, and enhance your problem sovling skills. This is an important exam. Many people will tell you that good letters are much more important than GRE scores, but the fact is that, if your GRE Sub score is not good enough, your file will not even be view thoroughly, so your letters may be ignored without reading.

Indeed, there are some people with really low GRE but admitting to really top schools, but you need to look deeply and will find that they alreayd have many good publications, so GRE, in this case, is indeed not that important. Below is my Summary of Knowledge and of Problems you will need for the preparation.

Measure (mathematics) - Wikipedia

But be sure to look for more problems to do, since those are really not enough. Those are hand-written, and I also suggest having a summary created on your own. Solutions Section 3 Section 7. Solution Section 9 Section Let's cheer for them! Measure Theory Midterm Review 1. Measure Theory Midterm Review Summary 2. Midterm Review Problems 3. Lebesgue Analysis 2. New results and facts given in the book are based on or closely connected with traditional topics of set theory, measure theory and general topology such as: infinite combinatorics, Martin's Axiom and the Continuum Hypothesis, Luzin and Sierpinski sets, universal measure zero sets, theorems on the existence of measurable selectors, regularity properties of Borel measures on metric spaces, and so on.

Essential information on these topics is also included in the text primarily, in the form of Appendixes or Exercises , which enables potential readers to understand the proofs and follow the constructions in full details. This not only allows the book to be used as a monograph but also as a course of lectures for students whose interests lie in set theory, real analysis, measure theory and general topology. Nonmeasurable sets and functions by A.

Historical background

B Kharazishvili 14 editions published in in English and held by WorldCat member libraries worldwide Taking as its starting point the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense, this book is devoted to various constructions of sets which are nonmeasurable with respect to invariant or quasi-invariant measures.

B Kharazishvili Book 6 editions published in in English and held by WorldCat member libraries worldwide "This book explores strange functions in real analysis and their applications. This book presents basic set-theoretical concepts such as binary relations of special type and the Generalized Continuum Hypothesis. It examines various functions whose constructions need essentially noneffective methods and those whose existence arises from known hypotheses.

It also discusses basic concepts of general topology and classical descriptive set theory. This second edition features five new chapters, with revised material throughout the text. It includes additional exercises as well as an expanded reference list. Strange Functions in Real Analysis is a valuable resource for students and mathematicians. Geometric aspects of probability theory and mathematical statistics by V.

V Buldygin Book 14 editions published between and in English and held by WorldCat member libraries worldwide This book demonstrates the usefulness of geometric methods in probability theory and mathematical statistics, and shows close relationships between these disciplines and convex analysis. Deep facts and statements from the theory of convex sets are discussed with their applications to various questions arising in probability theory, mathematical statistics, and the theory of stochastic processes.

The book is essentially self-contained, and the presentation of material is thorough in detail. Audience: The topics considered in the book are accessible to a wide audience of mathematicians, and graduate and postgraduate students, whose interests lie in probability theory and convex geometry.

Set theoretical aspects of real analysis by A. B Kharazishvili 12 editions published between and in English and held by WorldCat member libraries worldwide This book addresses a number of questions in real analysis and classical measure theory that are of a set-theoretic flavor. Accessible to graduate students, the beginning of the book presents introductory topics on real analysis and Lebesque measure theory.

These topics highlight the boundary between fundamental concepts of measurability and non-measurability for point sets and functions. The remainder of the book deals with more specialized material on set-theoretical real analysis.

Topics in Measure Theory and Real Analysis: The Measure Extension Problem and Related Questions
Topics in Measure Theory and Real Analysis: The Measure Extension Problem and Related Questions
Topics in Measure Theory and Real Analysis: The Measure Extension Problem and Related Questions
Topics in Measure Theory and Real Analysis: The Measure Extension Problem and Related Questions
Topics in Measure Theory and Real Analysis: The Measure Extension Problem and Related Questions

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