That supplement can serve as an overview of the elementary undergraduate real analysis used in this book. Chapter 2: This chapter begins by defining the outer measure on Ras a natural extension of the length function on intervals.
EXERCISES 1A
To say that f is Riemann integrable on [c,d] means that f is Riemann integrable with its domain restricted to [c,d].]. Prove that f is Riemann integrable on[a,b]if and only if is Riemann integrable on[a,c]and f is Riemann integrable on[c,b].
EXERCISES 1B
The size of As should be at most the sum of the lengths of a series of open intervals whose union contains A. The example from the paragraph above should provide intuition for the direction of the inequality in the following result.
EXERCISES 2A
We will show that µ has all the properties of external measure that were used in the proof of 2.18. We have shown that µ has all the properties of external measure that were used in the proof of 2.18.
EXERCISES 2B
Prove that if f(x)>0 for all x∈X, then fg(which is a function whose value at x∈X is equal to f(x)g(x)) is an S-measurable function. The hypothesis that µ(D) < ∞ is necessary in part (b) of the following result to avoid undefined terms of the form ∞−∞.
EXERCISES 2C
It is not known whether every number in the Cantor group is either rational or transcendental (meaning not a root of a polynomial with integer coefficients). However, many elements of the Cantor group are not endpoints of any interval in any Gn. Proof Every group used in the definition of the Cantor group is a union of open intervals.
Thenx ∈/ Cbecausex was removed in the first step of the Cantor set definition. Now we can use the Cantor function to show that the Cantor set is uncountable, even though it is a closed set with outer measure 0.
EXERCISES 2D
If A is a Lebesgue-measurable subset of R, then the above definition is the standard definition of an S-measurable function, where S is the σ-algebra of all Lebesgue-measurable subsets of A. A measurable Lebesgue set is an element of the smallest σ-algebra of R that contains all open subsets of Rand, and all subsets of R with outer measure 0. The outer measure restricted to the σ-algebra of measurable Lebesgue sets is also called Lebesgue measure.
A function f: A→R, where A⊂R, is called Lebesgue measurable if f−1(B) is a Lebesgue measurable set for every Borel set B⊂R. Similarly, a Lebesgue measurable function that is not Borel measurable is unlikely to appear in anything you do.
EXERCISES 2E
The next definition should remind you of the definition of the lowest Riemann sum (see 1.3). For the proof of the Monotone Convergence Theorem (and some other results), we will need to use the following soft restatement of the definition of the integral of a nonnegative function. 1∩A2, where the three groups appearing on the right-hand side of the above equation are disjoint.
Recall that integration with respect to a measure is defined via lower Lebesgue sums in the same way as the definition of the lower Riemann integral via lower Riemann sums (with the major exception of allowing measurable sets instead of just intervals in the partitions). If f ≥ 0, then f+ = f and f− =0; thus this definition is consistent with the previous definition of the integral of a non-negative function.
EXERCISES 3A
The following result could be proved as a special case of the dominant convergence theorem (3.31), which we prove later in this section. Therefore, you may want to read the simple proof of the bounded convergence theorem presented next. The bounded convergence theorem (3.26) requires that the measure of the entire space be finite and requires that the sequence of functions be uniformly bounded by a constant.
In the theorem below, the left side of the last equation denotes the Riemann integral. For example, the following result plays an important role in the proof of Lebesgue's differentiation theorem (4.10).
EXERCISES 3B
Is there a measurable Lebesgue group E⊂[0, 1], possibly constructed in a similar way to the Cantor group, such that the above equation is valid for allb∈[0, 1]. This tool is used to prove an almost ubiquitous version of the Fundamental Theorem of Calculus. The next result is a key tool in proving the maximum Hardy–Littlewood inequality (4.8).
In this example, I1,I4 is the only sublist of I1,I2,I3,I4 that yields the conclusion of the Vitali Covering Lemma. The maximum Hardy-Littlewood inequality proven in the following result is a key ingredient in the proof of Lebesgue's differentiation theorem (4.10).
EXERCISES 4A
The next result is called the Lebesgue Differentiation Theorem, although no derivative is in sight. Before coming to the formal proof of this first version of the Lebesgue Differentiation Theorem, we pause to provide a motivation for the proof. However, our next result says that everything is good almost everywhere, even in the absence of continuity in the function being integrated.
According to the first version of Lebesgue's differentiation theorem (4.10), the last quantity has limit 0ast → 0 for almost every b∈R. The following definition summarizes the idea of the proportion of a set in small intervals, centered on a numberb.
EXERCISES 4B
The next result shows that cross sections preserve measurability, this time in the context of functions rather than groups. Thus an algebra is closed under complement and under finite unions; a σ-algebra is closed under complementary and countable unions. Thus, the complement of a finite union of rectangles measurable inS ⊗ T is inA (use De Morgan's laws and the result in the previous paragraph that A is closed under finite intersections).
We now define a monotone class as a collection of groups that are closed under countably increasing unions and under countably decreasing intersections. However, some monotone classes are not closed even under finite unions, as shown by the following example.
EXERCISES 5A
See Exercise 1 of this section for an example (with finite measures) showing that Tonelli's theorem can fail without the hypothesis that the function being integrated is nonnegative. The following useful consequence of Tonelli's theorem states that we can switch the order of summation in a double sum of non-negative numbers. Fubini's theorem instead requires that the integral of the absolute value of the function be finite.
When using Fubini's theorem to evaluate the integral f, you will usually first use Tonelli's theorem applied to |f| to verify the hypothesis of Fubini's theorem. Tonelli's theorem (5.28) tells us that we can change the order of integration in the double integral above and get
EXERCISES 5B
The union of every set (finite or infinite) of open subsets of R is an open subset of Rn. The intersection of every finite set of open subsets of Rn is also an open subset of Rn. We can now prove that the product of two open sets is an open set.
To demonstrate the inclusion of the crowd in the other direction, temporarily fix the open crowd. In other words, we proved that if A∈ Band is an open subset of Rn, then A×G∈ Bm+n.
EXERCISES 5C
Specifically, we want the distance between two elements of our metric space to be a nonnegative number that is 0 if and only if the two elements are the same. Our next definition states that a subset of a metric space is open if every element in the subset is the center of an open ball contained in the subset. For example, every closed ballB(f,r) in a metric space is closed, as you are asked to prove in Exercise 3.
Limits on a metric space are defined by reducing to the context of the real numbers, where the limits are already defined. The combination of the two points in the result below shows that a subset of a complete metric space is complete if and only if it is closed. 6.16 the relation between complete and closed. a).
EXERCISES 6A
Definition F
Then, as you should verify, FX is a vector space; the additive identity in this vector space is the function0∈FX defined by 0(x) =0 for allx∈X. A subset of UofVis, called asubspaceofVifU, is also a vector space (using the same addition and scalar multiplication as onV). The following result gives the easiest way to check whether a subset of a vector space is a subspace.
Proof If is a subspace of V, then Satisfies the above three conditions by defining the vector space. SetiC([0, 1]) of continuous real-valued functions in[0, 1] is a vector space overRbecause the sum of two continuous functions is continuous and a constant multiple of a continuous function is continuous.
EXERCISES 6B
The next result shows that every normed vector space is also a metric space in a natural way. In a minor misuse of terminology, we often refer to a normed vector space V without mentioning the norm∥·∥. In other words, a normed vector space Show a Banach space if every Cauchy sequence inVconverges to an element of V.
The following result shows that if VandWare is a normed vector space, then B(V,W) is a normed vector space with the norm defined above. Verification of other properties required for a normed vector space is left to the reader.
EXERCISES 6C
The second point in Example 6.50 shows that there is a discontinuous linear functionality on a given normalized vector space. Our next major goal is to show that any infinite-dimensional normalized vector space has a discontinuous linear functionality (see 6.62). Now we can prove the promised result about the existence of discontinuous linear functionals on any infinite-dimensional normalized vector space.
In the last subsection, we showed that there exists a discontinuous linear functional on every infinite-dimensional normed vector space. Then the dual space ofV, denoted V′, is the normed vector space consisting of the bounded linear functionals ofV.
EXERCISES 6D
The following result gives another wonderful application of the Hahn–Banach theorem with the useful necessary and sufficient condition that an element of a normed vector space is in the closure of a subspace. 10 Give a concrete example of an infinite-dimensional normed vector space and the basis of this normed vector space. 16 Show that the dual space of every infinite-dimensional normed vector space is infinite-dimensional.
A normed vector space is called separable if it has a countable subset whose closure is equal to the entire space. The dual dual space of a normed vector space is defined as the dual space of the dual space.