The Classical Theory
Riemann Integration
If C is a nonoverlapping finite collection of rectangles, each contained in rectangle J, then vol (J)≥P. Furthermore, by Cauchy's convergence criterion, it is clear that bounded f is Riemann integrable if and only if.
Riemann–Stieltjes Integration
- Riemann Integrability
- Functions of Bounded Variation
Finally, if ψ has bounded variation in J, then every ϕ∈C(J;R) is Riemann integrable in J with respect to ψ. If every ϕ∈C(J;R) is Riemann integrable in J with respect to ψ, and if there is aK <∞ such.
Rate of Convergence
- Periodic Functions
- The Non-Periodic Case
It is not used in the proof of the next lemma, a result that is interesting in its own right. The following statement simply summarizes obvious applications of the results in §§3.1 and 3.2 to the present context.
Measures
Some Generalities
- The Idea
- Measures and Measure Spaces
After comparing Lebesgue's ideas with Riemann's in the countable setting, I conclude this introduction to Lebesgue's theory with a few words about the same equation for uncountable spaces. So, we now know that ¯µ is the only extension of µ as a measure on (E,Bµ) and that Bµ =B as (E,B, µ) is complete.
A Construction of Measures
- A Construction Procedure
- Lebesgue Measure on R N
- Distribution Functions and Measures
- Bernoulli Measure
- Bernoulli and Lebesgue Measures
Another advantage of (2.2.4) is that it facilitates the proof of the second part of the following preliminary addition result around ˜µ. One of the two key ingredients in Friedrichs' procedure is the following direct consequence of Lemma 6.3.10.
Lebesgue Integration
The Lebesgue Integral
- Some Miscellaneous Preliminaries
- The Space L 1 (µ; R )
It is then easy to see that (α, β)7−→α+β is continuous on Rc2ind in R, and therefore it is definitely a measurable map on Rc2,B. From a measure-theoretic point of view, the most elementary functions are those that take only a limited number of distinct values, and it is therefore reasonable for such a function to be said to be simple. Although this definition is obviously the correct one, it is not immediately obvious that it results in an integration where the integral is a linear function of the integrand.
Before accepting this definition, however, we must first check that it does not depend too much on the choice of the approximating sequence. Since it is clear that N(µ) is a linear subspace of L1(µ;R), we see that ∼µ is an equivalence relation and that the quotient space L1(µ;R)/ ∼µ is again a vector space over R .
Convergence of Integrals
- The Big Three Convergence Results
- Convergence in Measure
- Elementary Properties of L 1 (µ; R )
That is, convergence in µ-measure determines the limit function to exactly the same extent as µ-almost everywhere or k · kL1(µ;R)-convergence. It turns out that, after switching to a subseries, convergence in µ-measure leads to µ-convergence almost everywhere. Then there is a measurable function with R-value to which {fn : n≥1} converges in µ-measure if and only if.
Let (E,B, µ) be a finite measure space, and show that fn−→f in µ measure if and only ifR. Given an R-valued, measurable function f, show that (3.2.8) holds, and thus that fn −→f both (a.e.,µ) and inµ-measure if.
Lebesgue’s Differentiation Theorem
- The Sunrise Lemma
- The Absolutely Continuous Case
- The General Case
For us its importance is demonstrated in the following statement of the fundamental theorem of calculus. Lebesgue, distributions and polar coordinates The contents of this section consist of applications of the following general principle. Finally, to prove the last part of the theorem when µ(f > δ)<∞for any δ >0, just note that.
The essence of the relationship between these concepts and criterion theory is contained in the following. Convolutions and Young's inequality: One of the many applications of Theorem 6.3.2 is to the multiplication operation known as convolution.
Products of Measures
Fubini’s Theorem
But in that case f is the non-decreasing limit of non-negative measurable simple functions, and therefore, by the preceding and (d0),f ∈ K. The power of Lemma 4.1.1 to questions involving products is already clear in the following. Furthermore, by the same kind of argument as used to prove Lemma 4.1.2, for every non-negative measurable function f on (E1×E2,B1× B2), .
One may wonder why I separated the statement in Tonelli's Theorem from the statement in Fubini's Theorem. It is clear that (∗) can be interpreted as the statement that the integral of a non-negative function is the area under its graph.
Steiner Symmetrization
- The Isodiametric inequality
- Hausdorff’s Description of Lebesgue’s Measure
Therefore, after simple arithmetic manipulation and application of the monotone convergence theorem, we obtain Here are some simple applications of the previous ideas. i) Show that log is continuous and concave on every interval [∞) z > 0. In the second part I will look at mixed Lebesgue spaces, one of the many useful variants of the standard ones.
We now turn to the last part of the lemma, where both the measures µ1 and µ2 are assumed to be finite. More generally, if H is a Hilbert space over R, its complexification Hc is the complex vector space whose elements are of the formx+iy, wherex, y∈H.
Changes of Variable
Riemann vs. Lebesgue, Distributions, and Polar Coordinates
- Riemann vs. Lebesgue
- Polar Coordinates
Indeed, from the point of view of a probability purist, it is the distribution of a function, as opposed to the function itself, that is its characteristic. The reason it is sometimes useful to make this change of variables is that the integral on the right-hand side of (5.1.3) can often be evaluated as the limit of Riemann integrals, to which all the powerful calculus tools apply. From a theoretical point of view, the most interesting thing about this result is that it shows that a complete description of the class of Riemann integrable functions in terms of continuity properties defies a complete Riemannian solution and requires the Lebesgue concept almost everywhere.
Obviously, ψ is right-connected and non-increasing on [a, b], so (cf. exercise 2.1.17) B[a,b]-measurable there as well. Note: From now on, at least whenever the meaning is clear from the context, I will use the notation "dx" instead of the more cumbersome "λRN(dx)".
Jacobi’s Transformation and Surface Measure
- Jacobi’s Transformation Formula
- Surface Measure
If we start with the former, work first with simple functions and then move to monotone limits, we conclude that (5.2.3) holds for all non-negative, measurable functions f on Φ(G),BΦ(G). To begin with, let us say that M ⊂ RN is a hypersurface if for every p ∈ M there exist anr > 0 and three times continuous differentiable4 F : BRN(p, r)−→R with the properties that. Given a nonempty open set U in RN−1 and a three times continuously differentiable injection Ψ :U −→M with the property that.
Especially for the inverse function theorem, this means that for each u∈U there is a quarter of (u,0) in RN, where ˜Ψ is a diffeomorphism. Show that the tangent space Tp(M) for each p∈M coincides with the set of v∈RN such that.
The Divergence Theorem
- Flows Generated by Vector Fields
- Mass Transport
My first application of these considerations provides a generalization, known as Minkowski's inequality, of the triangle inequality. The following application, known as H¨older's inequality, gives a generalization of the inner product inequality|(ξ, η)RN| ≤ |ξ||η|forξ, η∈RN. The reader must verify that each of the statements here follows from the relevant result there.
Before applying H¨older's inequality to Lp spaces, it makes sense to complete the definition of the conjugate H¨older0 given in Theorem 6.1.5 only for p∈(1,∞). Much of the analysis depends on the clever selection of the "right" orthonormal basis for a given task.
Basic Inequalities and Lebesgue Spaces
Jensen, Minkowski, and H¨ older
Because it is a particularly significant case, it is often referred to by another name and is called Schwarz's inequality. Note the similarity between the development here and that of the classical triangle inequality for the Euclidean metric in RN. Fdµ∈C. Finally, let g :C −→[0,∞) be a continuous, concave function and use the first part to prove Jensen's inequality.
Here are some steps you might want to follow to prove Jensen's inequality. i) Show that if g1ogg2 are continuous, concave functions on C, then so is g1∧g2. In particular, ifg is a non-negative, continuous, concave function, so is g∧nis, and use this to reduce the proof of Jensen's inequality to the case where g is bounded.
The Lebesgue Spaces
- The L p -Spaces
- Mixed Lebesgue Spaces
In particular, if µ is σ-finite and B is generated by a countable collection C, then each of the spaces Lp(µ;R),p∈[1,∞), is separable. Proof: Without loss of generality, we will assume that all thefn's as well as gare are non-negative. The goal of this subsection is to show that when p1≤p2, the L(p1,p2) norm of a function is dominated by its L(p2,p1) norm, and the following lemma will play a crucial role in the proof.
Proof: Since it is easy to reduce the general case to the one in which both µ1 and µ2 are finite, we will take them as probability measures from the outset. Hint: First reduce to the case where µ1 and µ2 are finite, then apply part (iv) of Theorem 6.2.1 to handle this case.
Some Elementary Transformations on Lebesgue Spaces
- A General Estimate for Linear Transformations
- Convolutions and Young’s inequality
- Friedrichs Mollifiers
To understand how useful bump functions can be, remember the statement I gave about the divergence theorem. For fuller accounts the reader should consult one of the many excellent books on the subject. Assume that {en : n≥0} is an orthonormal sequence in H. Then, depending on whether H is real or complex, for each {αm: m≥0} ∈. Finally, the closed linear span1 Lof{en: n≥0}in H coincides with the set of sumP∞. 1 The closed linear span of a set is the closure of the subspace spanned by this set.
It may be reassuring to know that the dimension of a separable, infinite-dimensional Hilbert space is in some sense well-defined and equal to the cardinality of the integers. For example, you can change locations and replace [0,1) with the unit circle S1, which is considered a subset of the complex plane.
Hilbert Space and Elements of Fourier Analysis
Hilbert Space
- Elementary Theory of Hilbert Spaces
- Orthogonal Projection and Bases
That is, a Hilbert space overC is a vector space overC that has a Hermitian inner product (property (a) below) (z, ζ) ∈H2 7−→ (z, ζ)H ∈C with the properties that. If S is a subset of a real or complex Hilbert space H, then S contains a dense subset of H if and only if S⊥={0}. Let H be a real or complex Hilbert space and note that every closed subspace of H becomes a Hilbert space with the inner product obtained by the constraint.
Given a complex, infinite-dimensional, separable Hilbert space H, there are numerous ways to produce a real Hilbert space of which H is the complexification. Show that Li is also a separable Hilbert space and that any orthonormal basis for L can be extended to an orthonormal basis for H.
Fourier Series
- The Fourier Basis
- An Application to Euler–Maclaurin
Show that L becomes a true Hilbert space when one takes the restriction of (·,·)H to L to be its inner product. Show that Π is the orthogonal projection operator on the closed subspace Lif and only if L= Range(Π) and Π is idempotent and symmetric (i.e. Π2= Π and (Πx, y)H= (x,Πy)H for allx, y∈H). Also show that if L is a closed, linear subspace of H, then ΠL⊥ =I−ΠL, where I is the identity map.
To see this, show that if E is an infinite subset of H that is orthonormal in the sense that. That is, if E is an orthonormal basis for L, then there exists an orthonormal basis ˜E for H with E⊆E.
The Fourier Transform
- L 1 -Theory of the Fourier Transform
- The Hermite Functions
- L 2 -Theory of the Fourier Transform
The Radon–Nikodym Theorem, Daniell Integration,
The Radon–Nikodym Theorem
The Daniell Integral
- Extending an Integration Theory
- Identification of the Measure
- An Extension Theorem
- Another Riesz Representation Theorem
Carath´ eodory’s Method
- Outer Measures and Measurability
- Carath´ eodory’s Criterion
- Hausdorff Measures
- Hausdorff Measure and Surface Measure
