# 7.9.E: Problems on Lebesgue-Stieltjes Measures

Exercise (PageIndex{1})

Do Problems 7 and 8 in §4 and Problem 3' in §5, if not done before.

Exercise (PageIndex{2})

Prove in detail Theorems 1 to 3 in §8 for LS measures and outer measures.

Exercise (PageIndex{3})

Do Problem 2 in §8 for LS-outer measures in (E^{1}).

Exercise (PageIndex{4})

Prove that (f : E^{1} ightarrow(S, ho)) is right (left) continuous at (p) iff
[lim _{n ightarrow infty} fleft(x_{n} ight)=f(p) ext { as } x_{n} searrow pleft(x_{n} earrow p ight).]
[Hint: Modify the proof of Theorem 1 in Chapter 4, §2.]

Exercise (PageIndex{5})

Fill in all proof details in Theorem 2.
[Hint: Use Problem 4.]

Exercise (PageIndex{6})

In Problem 8(iv) of §4, describe (m_{alpha}^{*}) and (M_{alpha}^{*}).

Exercise (PageIndex{7})

Show that if (alpha=c)constant on an open interval (I subseteq E^{1}) then
[(forall A subseteq I) quad m_{alpha}^{*}(A)=0.]
Disprove it for nonopen intervals (I) (give a counterexample).

Exercise (PageIndex{8})

Let (m^{prime}: mathcal{M} ightarrow E^{*}) be a topological, translation-invariant measure in (E^{1}), with (m^{prime}(0,1]=c(i) (m^{prime}=c m) on the Borel field (mathcal{B}.) (Here (m: mathcal{M}^{*} ightarrow E^{*}) is Lebesgue measure in (E^{1}).)
*(ii) If (m^{prime}) is also complete, then (m^{prime}=c m) on (mathcal{M}^{*}).
(iii) If (0*(iv) If (mathcal{M}^{prime}=mathcal{B},) then (c m) is the completion of (m^{prime}) (Problem 15 in §6).
[Outline: (i) By additivity and translation invariance,
[m^{prime}(0, r]=c m(0, r]]
for rational
[r=frac{n}{k}, quad n, k in N]
(first take (r=n,) then (r=frac{1}{k},) then (r=frac{n}{k})).
By right continuity (Theorem 2 in §4), prove it for real (r>0) (take rationals (r_{i} searrow r)).
By translation, (m^{prime}=c m) on half-open intervals. Proceed as in Problem 13 of §8.
(iii) See Problems 4 to 6 in §8. Note that, by Theorem 2, one may assume (m^{prime}=m_{alpha}) (a translation-invariant (L S) measure). As (m_{alpha}=c m) on half-open intervals, Lemma 2 in §2 yields (m_{alpha}=c m) on (mathcal{G}) (open sets). Use (mathcal{G})-regularity to prove (m_{alpha}^{*}=c m^{*}) and (mathcal{M}_{alpha}^{*}=mathcal{M}^{*}).]

Exercise (PageIndex{9*})

(LS measures in (E^{n}.)) Let
[mathcal{C}^{*}=left{ ext {alf-open intervals in } E^{n} ight}.]
For any (operatorname{map} G : E^{n} ightarrow E^{1}) and any ((overline{a}, overline{b}] in mathcal{C}^{*},) set
[egin{aligned} Delta_{k} G(overline{a}, overline{b}] &=Gleft(x_{1}, ldots, x_{k-1}, b_{k}, x_{k+1}, ldots, x_{n} ight) &-Gleft(x_{1}, ldots, x_{k-1}, a_{k}, x_{k+1}, ldots, x_{n} ight), quad 1 leq k leq n. end{aligned}]
Given (alpha : E^{n} ightarrow E^{1},) set
[s_{alpha}(overline{a}, overline{b}]=Delta_{1}left(Delta_{2}left(cdotsleft(Delta_{n} alpha(overline{a}, overline{b}] ight) cdots ight) ight).]
For example, in (E^{2}),
[s_{alpha}(a, b]=alphaleft(b_{1}, b_{2} ight)-alphaleft(b_{1}, a_{2} ight)-left[alphaleft(a_{1}, b_{2} ight)-alphaleft(a_{1}, a_{2} ight) ight].]
Show that (s_{alpha}) is additive on (mathcal{C}^{*}). Check that the order in which the (Delta_{k}) are applied is immaterial. Set up a formula for (s_{alpha}) in (E^{3}).
[Hint: First take two disjoint intervals
[(overline{a}, overline{q}] cup(overline{p}, overline{b}]=(overline{a}, overline{b}],]
as in Figure 2 in Chapter 3, §7. Then use induction, as in Problem 9 of Chapter 3, §7.]

Exercise (PageIndex{10*})

If (s_{alpha}) in Problem 9 is nonnegative, and (alpha) is right continuous in each variable (x_{k}) separately, we call (alpha) a distribution function, and (s_{alpha}) is called the (alpha)-induced (L S) premeasure in (E^{n};) the (L S) outer measure (m_{alpha}^{*}) and measure
[m_{alpha} : mathcal{M}_{alpha}^{*} ightarrow E^{*}]
in (E^{n}) (obtained from (s_{alpha}) as shown in } §§5 and 6) are said to be induced by (alpha.)
For (s_{alpha}, m_{alpha}^{*},) and (m_{alpha}) so defined, redo Problems 1-3 above.

## Stieltjes

Nicolae Dinculeanu , in Handbook of Measure Theory , 2002

### 6.4. The Stieltjes integral

Let g: ℝ → EL(F, G) be a function and let mg: R → E be the finitely additive measure associated to g.

We define first the Stieltjes integral ∫ f dg in case g is right continuous and has finite variation function |g|. In this case, by Theorem 23 , the measure mg can be extended to a σ-additive measure m: D → E with finite variation |m|. We shall denote m still by mg we have

We can consider the space L F 1 ( m g ) = L F 1 ( | m g | ) , in the sense of stage 3 of the development of the integral.

We shall denote L F 1 ( m g ) by L F 1 ( g ) For every f ∈ L F 1 ( g ) we define the Lebesgue-Stieltjes integral ∫ f dg by the equality

If f ∈ L F 1 ( m g ) then | f | ∈ L 1 ( m | g | ) and we have

We consider now the case where g has finite semivariation function ( g ˜ ) ℝ . E and there is a space ZG * norming for G ** (for example Z = G * ), such that for each zZ, the function gz: ℝ → F * is right continuous. Then, by Theorem 24 , mg can be extended to an additive measure m g : D → L ( F , Z * ) with finite semivariation ( m g ˜ ) F . Z * such that for each zZ, the measure (mg)z is σ-additive. We can consider then the space ℱ F . Z * ( m g ) defined in Section 5 . We denote F F . Z * ( m g ) by F F . Z * ( g ) and for every function f ∈ F F . Z * ( g ) we define the Lebesgue-Stieltjes integral ∫fdg by the equality

If, in addition, c o ⊄ G , then

Assume g is right continuous and has finite variation function |g|. Then g has also finite semivariation function g ˜ F . G relative to any embedding EL(F, G). The measure mg has finite variation |mg| = m|g| and finite semivariation ( m ˜ g ) F . G . We have

For f ∈ L F 1 ( m g ) the Stieltjes integral ∫ f dg is the same, whether we consider f in L F 1 ( m g ) or in F F . G ( m g ) .

We could consider the semiring P ′ of the intervals of the form [a, b[and the ring R ′ generated by P ′ and define the measure m ′ g : R ′ → E by.

Then m ′ g is σ-additive and has finite variation (respectively finite semivariation) iff g is left continuous and has finite variation (respectively finite semivariation). If m ′ g has finite variation, then m ′ g can be extended to a σ-additive measure m ' with finite variation on the δ-ring D. If m has finite semivariation and if for every z ∈ Z, the function gz is left continuous, then m ′ g can be extended to an additive measure m with finite semivariation on D, such that for every zZ, the measure m ′ z . is σ-additive.

If we start from a function g: ℝ → E with finite variation but not necessarily left or right continuous, it is more appropriate to define the measure m: R → E by

and the measure m ′ : R → E by

Then both measures are σ-additive and can be extended to the same σ-additive measure on D. In fact, the function g+(t) = g(t+) is right continuous and the function g−(t) = g(t−) is left continuous and we have

Similar considerations can be made in case g has finite semivariation. But in this case, g does not necessarily have lateral limits in G. However, there are elements G(t+) and g(t−) in G ** such that for every xF and z ∈G * we have.

## Measures of Variability

A series of free, online video lessons with examples and solutions to help Grade 7 students learn how to informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team on a dot plot, the separation between the two distributions of heights is noticeable.

### Suggested Learning Targets

• I can calculate the mean, range, and the mean absolute deviation (MAD) to compare two data sets (Note: MAD is the average distance between each value and the mean.)
• I can observe the overlap and differences of two data sets with similar variability.
• I can compare two data sets using the range or MAD. Variability and Deviations from the Mean
Summarizing Deviations from the Mean.
&bull Variability describes how spread out the data is.
&bull For any given value in a data set, the deviation from the mean is the value minus the mean.
&bull The greater the variability (spread) of the distribution, the greater the deviations from the mean (ignoring the signs of the deviation).

Mean Absolute Deviation
This video reviews how to find Mean Absolute Deviation for a set of data.
One way to find out how consistent a set of data is to find the Mean Absolute Deviation. The Mean Absolute Deviation describes the average distance from the mean for the numbers in the data set.
Step 1: Find the mean of the data.
Step 2: Subtract the mean from each data point. (Make all values positive)
Step 3: Find the mean of the values you got when you subtracted in step 2. Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. ## Intuition

The Lebesgue integral works by calculating the value of an integral based on y y y -values instead of x x x -values. Let

f ( x ) = < 1 4 if 0 ≤ x ≤ 3 4 1 2 if 3 4 < x ≤ 1. f(x)=egin frac<1> <4> ext < if >0leq xleq frac<3><4> frac<1><2> ext < if >frac<3><4><xleq 1. end f ( x ) = ⎩ ⎪ ⎨ ⎪ ⎧ ​ 4 1 ​ if 0 ≤ x ≤ 4 3 ​ 2 1 ​ if 4 3 ​ < x ≤ 1 . ​

What is the value of ∫ 0 1 f ( x ) d x int_0^1 f(x), dx ∫ 0 1 ​ f ( x ) d x ?

This graph consists of two line segments, so the area under it can be thought of as two rectangles, so the integral has value 3 4 ⋅ 1 4 + 1 4 ⋅ 1 2 = 5 16 . frac<3><4>cdot frac<1><4>+frac<1><4>cdot frac<1><2>=frac<5><16>. 4 3 ​ ⋅ 4 1 ​ + 4 1 ​ ⋅ 2 1 ​ = 1 6 5 ​ . When we use the Riemann integral though, we're actually thinking about this slightly differently: we're drawing many smaller rectangles, and using them to "approximate" the large rectangles, although in this case the approximation is exact.

The Lebesgue integral thinks about this problem in a different way: the function f f f takes only the values 1 4 frac14 4 1 ​ and 1 2 frac12 2 1 ​ , so we consider the size of the sets on which f f f takes those values. They are 3 4 frac34 4 3 ​ and 1 4 frac14 4 1 ​ respectively, so the total area must be 1 4 ⋅ 3 4 + 1 2 ⋅ 1 4 = 5 16 . □ frac<1><4>cdot frac<3><4>+frac<1><2>cdot frac<1><4>=frac<5><16>. _square 4 1 ​ ⋅ 4 3 ​ + 2 1 ​ ⋅ 4 1 ​ = 1 6 5 ​ . □ ​

In this case, the distinction between the two ways of thinking about the area is meaningless, but as the following example shows, this is not always the case.

In essence, the Lebesgue integral is looking at how often a function achieves a certain value rather than the value of a function at a particular point. According to Reinhard Siegmund-Schultze  , Lebesgue himself explained this idea in a letter to Paul Montel, writing

"I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral."

If the above equation is true for coprime positive integers a a a and b b b , find a + b a+b a + b .

## 3 Main results

The solutions of usual differential problems are at least continuous, but if we look at, for example, a big part of the class of hybrid systems, this is not available. In our problems, by considering inclusions driven by general Borel measures, it cannot be expected to obtain continuous solutions. This is the reason for which in the following definition the using of the left limit:

is crucial. Indeed, as pointed out in an example in , taking the integral on a closed, resp. open interval (which is equivalent to integrating on a closed interval the left limit of the function) leads to completely different solutions.

Definition 4 A solution of the problem (1) is a function x : [ 0 , 1 ] → R d for which there exists a μ-integrable function g : [ 0 , 1 ] → R d such that

Let us note that for discontinuous solutions there is always a problem how to define a solution in the points of discontinuity. A very interesting survey on the topic can be found in . We need to remark that distinct definitions of solutions can lead to distinct solutions ! The existence and uniqueness of solutions of considered problems depend on the conditions for μ and g. Due to the existence of atoms for the measure μ there is a question about the uniqueness of solutions for a (possibly) discontinuous function g. The explicit scheme Δ x ( t ) = x ( t ) − x ( t − ) = g ( x ( t − ) ) is a natural choice for physical systems and we will follow this idea. This allows us to fill the gap in this theory. If we compare our result with some earlier ones we need to recall that for purely atomic measure μ the above condition of integrability means that the series ∑ k g ( t k ) μ < t k >is finite (where t k is a set of atoms for μ) - cf. [8, 16], for instance.

Let us present some auxiliary result:

Lemma 5 Let Γ : R d → P c c ( R d ) be an usc multifunction and ( x n ) n be a sequence that converges to x ∈ R d . Suppose that there exists a constant M > 0 such that d ( 0 , Γ ( x n ) ) ≤ M for every n ∈ N . Then

Proof By the upper semicontinuity of Γ it follows that d ( 0 , Γ ( ⋅ ) ) is lower semicontinuous (see [, Lemma 9.3.1]). Therefore

We are ready to present our first result for measure-driven differential inclusions (1) for a general class of finite Borel measures. Let us note that the presented theorem is intended to unify and to extend the earlier ones. We not only formulate an existence result, but we also include a method how to find this solution as a limit of some approximations. We refer the reader to  for the discussion, some motivations and examples for measure-driven problems.

Theorem 6 Let μ be a finite Borel measure on [ 0 , 1 ] and let G : [ 0 , 1 ] × R d → P c c ( R d ) satisfy the following hypotheses:

G ( ⋅ , ⋅ ) is product Borel measurable,

G ( t , ⋅ ) is usc for every t ∈ [ 0 , 1 ] ,

there exists a μ-integrable function M : [ 0 , 1 ] → R + such that

Then there exists at least one solution for the measure-driven differential problem (1).

Proof Our proof is based on an iteration procedure. More precisely, we construct a sequence of approximate solutions (being regulated functions) which is shown to have a convergent subsequence due to some compactness properties.

So, let x 0 ( t ) = x 0 for t ∈ [ 0 , 1 ] . Suppose then that we have already constructed a regulated (bounded variation (BV)) function x n on [ 0 , 1 ] and choose x n + 1 by following a scheme that is described in the sequel.

By our hypotheses on G we ensure that the function G is superpositionally Borel measurable . Since x n is regulated (BV), there exists x n ( t − ) at every point t. It can be obtained as x n ( t − ) = lim m → ∞ x n ( t − τ m ) , where ( τ m ) m is a sequence of positive numbers converging to 0 (such that t − τ m ∈ [ 0 , 1 ] ). Therefore the function t ↦ x n ( t − ) is measurable as a pointwise limit of a sequence of measurable functions and then the multifunction t ↦ G ( t , x n ( t − ) ) is Borel measurable too.

By Theorem 12.1 in  (cf. also Chapter III in ) it follows that t ↦ d ( 0 , G ( t , x n ( t − ) ) ) is Borel measurable. Moreover, by hypothesis (3) and Lemma 5, it is bounded by M ( t ) . Since the values of G are closed and convex we are able to find a Borel measurable selection g n ( ⋅ ) of G ( ⋅ , x n ( ⋅ − ) ) such that

(in our finite-dimensional case, it is unique). Define now

As it was presented in the preliminary part of the paper (Theorem 1), the measure μ is, in fact, a Lebesgue-Stieltjes measure with respect to a BV, right-continuous function F. Thus the previous integral should be understood in the sense of ∫ 0 t g n ( s ) d μ F ( s ) i.e. as a Lebesgue-Stieltjes integral. This integral is well defined since the selection g n is Borel measurable, F is of bounded variation and, as said before, bounded by M ( t ) . Moreover, by Proposition 3, x n + 1 is a regulated function.

The function F is of bounded variation and right-continuous, therefore it has at most countable points of left discontinuity. Let A = < t k : k ∈ N >be the set of its discontinuity points.

As the sequence ( g n ) n is pointwise bounded, we can extract, by a diagonal procedure, a subsequence (not re-labeled) for which

As a pointwise limit of measurable functions g ˜ is measurable on A. By (3), using the Lebesgue dominated convergence result Theorem 5.3.3 in , we obtain for every t ∈ [ 0 , 1 ]

At the same time, the sequence ( g n χ [ 0 , 1 ] ∖ A ) n is uniformly integrable in L 1 ( [ 0 , 1 ] ∖ A , μ F ) and bounded. Whence it is relatively weakly compact in the space L 1 ( [ 0 , 1 ] ∖ A , μ F ) (cf. ). It follows that we can extract a subsequence (denoted in the same way, for the sake of convenience) which converges in the weak topology of L 1 ( [ 0 , 1 ] ∖ A , μ F ) to some function h ∈ L 1 ( [ 0 , 1 ] ∖ A , μ F ) . In particular,

Gurman, V.I., Vyrozhdennye zadachi optimal’nogo upravleniya (Degenerate Problems of Optimal Control), Moscow: Nauka, 1977.

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Miller, B.M. and Rubinovich, E.Ya., Impulsive Control in Continuous and Discrete-Continuous Systems, New York: Kluwer, 2003. Translated under the title Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami, Moscow: Nauka, 2005.

Zavalishchin, S.T. and Sesekin, A.N., Impul’snye protsessy: modeli i prilozheniya, Moscow: Nauka, 1991. Translated under the title Dynamic Impulse Systems: Theory and Applications, Dorderecht: Kluwer, 1997.

Dykhta, V.A. and Samsonyuk, O.N., Optimal’noe impul’snoe upravlenie s prilozheniyami (Optimal Impulsive Control with Appendices), Moscow: Fizmatlit, 2000.

Goncharova, E.V. and Staritsyn, M.V., Optimal Impulsive Control Problems with State and Mixed Constraints, Dokl. Math., 2011, vol. 84, no. 3, pp. 882–885.

Goncharova, E. and Staritsyn, M., Optimization of Measure-driven Hybrid Systems, J. Optim. Theory Appl., 2012, vol. 153, no. 1, pp. 139–156.

Arutyunov, A.V., Karamzin, D.Yu., and Pereira, F., On Constrained Impulsive Control Problems, J. Math. Sci., 2010, vol. 165, pp. 654–688.

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Miller, B. and Bentsman, J., Optimal Control Problems in Hybrid Systems with Active Singularities, Nonlin. Anal., 2006, vol. 65, no. 5, pp. 999–1017.

Boccadoro, M., Wardi, Y., Egerstedt, M., and Verriest, E., Optimal Control of Switching Surfaces in Hybrid Dynamical Systems, Discrete Event Dyn. Syst.: Theory Appl., 2005, vol. 15, no. 4, pp. 433–448.

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Dykhta, V.A., Variatsionnyi printsip maksimuma i kvadratichnye usloviya optimal’nosti impul’snykh i osobykh rezhimov (Variational Principle of Maximum and Quadratic Conditions for Optimality of Impulsive and Special Modes), Irkutsk: Irkutsk. Gos. Ekonom. Akad., 1995.

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## 9 Ways To Deal With Difficult Employees

Nearly every manager I’ve ever consulted to or coached has told me about having at least one employee who’s not so great. I’ve come to think of it as an almost inevitable part of the manager’s professional landscape: there's generally that one (or more) employee who doesn’t perform well, or is difficult to deal with, or has a hard time getting along with others, or means well but just doesn’t ever quite do what’s expected, or….

And the unfortunate thing is, most managers get held hostage to these folks, spending a disproportionate amount of time, thought and emotional energy on them. Often hovering on the verge of letting them go for years, but never quite being able (for a variety of reasons) to pull the trigger.

Here, then, are nine things that excellent managers do when confronted with a difficult employee – things that keep them from getting sucked into an endless vortex of ineffectiveness and frustration:

Listen. Often, when an employee is difficult we stop paying attention to what’s actually going on. We're irritated, it seems hopeless, and we’ve already decided what we think about the employee - so we just turn our attention to other things, out of a combination of avoidance and self-protection. But the best managers get very attentive when someone’s not doing well. They know their best shot at improving the situation lies in having the clearest possible understanding of the situation – including knowing the tough employee’s point of view. An added bonus: in some cases, simply listening can save the day. You may hear about a real problem that’s not the employee’s fault that you can solve the tough employee may start acting very differently once he or she feels heard you may discover legitimate issues he or she has that need to be addressed.

If you learn to use these ‘good manager’ approaches when you have a difficult employee, then no matter how things turn out, you’ll end up knowing that you’ve done your best in a tough situation. And that may be the best stress reducer of all.

Check out Erika Andersen’s latest book, Leading So People Will Follow,and discover how to be a followable leader. Booklist called it “a book to read more than once and to consult many times.”

## Construction of Measures

Construction of measures in this chapter follows the approach of C. Carathéodory, which is based on the concept of outer measure and has the advantage of introducing measurable sets solely in terms of outer measure this is a natural way of constructing measures, and provides feasible ways to look for appropriate measure spaces when a suitable choice of measure space is at issue for the study of a given problem. Usually a certain regularity property of outer measures is desirable in this undertaking the Carathéodory approach proves to be most handy in this respect. The Lebesgue–Stieltjes measure, constructed from a given increasing function, illustrates this point clearly. The Lebesgue measure on Euclidean n-space is introduced separately for n = 1 and n > 1 in this chapter, followed by some exercises which are designed to show to the reader the power gained by using integral based on the Lebesgue measure.

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

## Relationship Problem: Trust

Trust is a key part of a relationship. Do you see certain things that cause you not to trust your partner? Or do you have unresolved issues that prevent you from trusting others?

Problem-solving strategies:

You and your partner can develop trust in each other by following these tips, Fay says.

• Be consistent.
• Be on time.
• Do what you say you will do.
• Don't lie -- not even little white lies to your partner or to others.
• Be fair, even in an argument.
• Be sensitive to the other's feelings. You can still disagree, but don't discount how your partner is feeling.
• Call when you say you will.
• Call to say you'll be home late.
• Don't overreact when things go wrong.
• Never say things you can't take back.
• Don't dig up old wounds.
• Don’t be jealous.
• Be a good listener.

Even though there are always going to be problems in a relationship, Sherman says you both can do things to minimize marriage problems, if not avoid them altogether.

First, be realistic. Thinking your mate will meet all your needs -- and will be able to figure them out without your asking -- is a Hollywood fantasy. "Ask for what you need directly," she says.

Next, use humor -- learn to let things go and enjoy one another more.

Finally, be willing to work on your relationship and to truly look at what needs to be done. Don't think that things would be better with someone else. Unless you address problems, the same lack of skills that get in the way now will still be there and still cause problems no matter what relationship you're in.

#### Sources

Mary Jo Fay, RN, MSN, author, When Your "Perfect Partner" Goes Perfectly Wrong, Out of the Boxx, 2004 and Please Dear, Not Tonight, Out of the Boxx, 2006.

Karen Sherman, PhD, author, Marriage Magic! Find It, Keep It, and Make It Last. Dr. Karen Sherman, 2008.

Allison Cohen, MFT, psychotherapist, California.

Mitch Temple, author of The Marriage Turnaround, Moody Publishers, 2009.

Paulette Kouffman Sherman, PhD, author, Dating from the Inside Out: How to Use the Law of Attraction in Matters of the Heart, Atria Books/Beyond Words, 2008.

Gail Cunningham, spokeswoman, National Foundation for Credit Counseling.

Elaine Fantle Shimberg, author, Blending Families. Blending Families, 1999.

## 7.9.E: Problems on Lebesgue-Stieltjes Measures This book presents measure and integration theory in a self­contained and step by step manner. After an informal introduction to the subject, the general extension theorem of Caratheodory is presented in Chapter 1. This is followed by the construction of Lebesgue­Stieltjes measures on the real line and Euclidean spaces, and of measures on finite and countable spaces. The presentation gives a general perspective to the subject so as to enable students to think beyond the special, albeit important, example of the Lebesgue measure on the real line. Integration theory is developed in Chapter 2 where the three basic convergence theorems and their extensions are presented. Basic aspects of the theory of Lp, Banach and Hilbert spaces are presented in chapter 3. The Lebesgue­ Radon­Nikodym theorem, signed measures and the fundamental theorem of the Lebesgue integral calculus are taken up in chapter 4. Product measures and their applications to convolutions are discussed in Chapter 5. Also included in Chapter 5 are sections on Fourier Series and Fourier transforms. The last chapter is devoted to basic aspects of probability theory including the Kolmogorov consistency theorem for the construction of stochastic processes. The appendix reviews basic set theory and advanced calculus.

Apart from providing a modern presentation of the subject, this book includes a large number of exercises that should prove very useful to the instructor and the students. This book should be valuable to M.Sc. and Ph.D. students in mathematics, statistics and related fields in India.

1. Measures and Integration: An Informal Introduction
2. Measures
3. Integration
4. L p ­Spaces
5. Differentiation
6. Product Measures, Convolutions, and Transforms
7. Probability Spaces
A.1 Elementary set theory
A.2 Real numbers, continuity, differentiability and integration
A.3 Complex numbers, exponential and trigonometric functions
A.4 Metric spaces
A.5 Problems
List of Symbols and Abbreviations
References
Subject Index