The Lebesgue theory

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Chapter 11: The Lebesgue theory

This chapter will be very concise.

It is the purpose of this chapter to present the fundamental concepts of the Lebesgue theory of measure and integration and to prove some of the crucial theorems in a rather general setting, without obscuring the main lines of the development by a mass of comparatively trivial detail. Therefore proofs are only sketched or omitted in some cases.

The theory of the Lebesgue integral can be developed in several distinct ways. Only one of these methods will be discussed here. For alternative procedures we refer to the more specialized treatises on integration.

Contents
Contents
 1.  Set functions
 2.  Construction of the Lebesgue measure
 3.  Measure spaces
 4.  Measurable functions
 5.  Simple functions
 6.  Integration
 7.  Comparison with the Riemann integral
 8.  Integration of complex functions
 9.  Functions of class \mathcal{L}^2

1. Set functions If A and B are any two sets, we write A - B for the set of all elements x such that x \in A, x\notin B. The notation A- B does not imply that B\subset A. We denote the empty set by 0, and say that A and B are disjoint if A \cap B = 0.

We shall always assume that the range of \phi does not contain both +\infty and -\infty; for if it did, the right side of the additivity formula could become meaningless. Also, we exclude set functions whose only value is +\infty or -\infty.

It is interesting to note that the left side of the countable additivity formula is independent of the order in which the A_n's are arranged. Hence the rearrangement theorem shows that the right side of it converges absolutely if it converges at all; if it does not converge, the partial sums tend to +\infty, or to -\infty.

If \phi is additive, the following properties are easily verified:
\begin{gather*}
\phi(0)=0,\\
\phi(A_1\cup\dots\cup A_n)=\phi(A_1)+\dots+\phi(A_n) \quad\text{if } A_i\cap A_j=0 \text{ whenever } i\neq j,\\
\phi(A_1\cup A_2)+\phi(A_1\cap A_2)=\phi(A_1)+\phi(A_2).
\end{gather*}

If \phi(A)\geqslant0 for all A, and A_1\subset A_2, then \phi(A_1)\leqslant \phi(A_2). Because of this, nonnegative additive set functions are often called monotonic.

\phi(A-B)=\phi(A)-\phi(B) if B\subset A and |\phi B|<+\infty.

At this point, the following properties should be verified:

• \mathcal{E} is a ring, but not a \sigma-ring.
  
• If A\in\mathcal{E}, then A is the union of a finite number of disjoint intervals.
  
• If A\in\mathcal{E}, m(A) is well defined; that is, if two different decompositions of A into disjoint intervals are used, each gives rise to the same value of m(A).
  
• m is additive on \mathcal{E}.

Our next objective is to show that every regular set function on \mathcal{E} can be extended to a countably additive set function on a \sigma-ring which contains \mathcal{E}.

It is clear that \mu^\ast (E)\geqslant0 for all E and that \mu^\ast (E_1)\leqslant\mu^\ast (E_2) if E_1\subset E_2.

Note that The first one asserts that \mu^\ast is an extension of \mu from \mathcal{E} to the family of all subsets of \mathbb{R}^p. The second one is called subadditivity.

S(A, B) is the so-called "symmetric difference" of A and B. We shall see that d(A, B) is essentially a distance function.

The following theorem will enable us to obtain the desired extension of \mu.

We develop some of the properties of S(A, B) and d(A, B). We have
\begin{gather*}
S(A,B)=S(B,A),\quad S(A,A)=0, \\
S(A,B)\subset S(A,C)\cup S(C,B).\\
\left.\begin{array}{l}
S(A_1\cup A_2,B_1\cup B_2) \\
S(A_1\cap A_2,B_1\cap B_2) \\
S(A_1- A_2,B_1- B_2)
\end{array}\right\}\subset S(A_1,B_1)\cup S(A_2,B_2).
\end{gather*}

These properties of S(A,B) imply
\begin{gather*}
d(A,B)=d(B,A),\quad d(A,A)=0, \\
d(A,B)\leqslant d(A,C)+d(C,B).\\
\left.\begin{array}{l}
d(A_1\cup A_2,B_1\cup B_2) \\
d(A_1\cap A_2,B_1\cap B_2) \\
d(A_1- A_2,B_1- B_2)
\end{array}\right\}\leqslant d(A_1,B_1)+d(A_2,B_2).
\end{gather*}

If we define two sets A and B to be equivalent, provided d(A,B)=0, we divide the subsets of \mathbb{R}^p into equivalence classes, and d(A, B) makes the set of these equivalence classes into a metric space. \mathfrak{M}_F(\mu) is then obtained as the closure of \mathcal{E}.

one more property of d(A, B): |\mu^\ast (A)-\mu^\ast (B)|\leqslant d(A,B), if at least one of \mu^\ast (A),\mu^\ast (B) is finite.

We now replace \mu^\ast (A) by \mu(A) if A\in \mathfrak{M}(\mu). Thus \mu, originally only defined on \mathcal{E}, is extended to a countably additive set function on the \sigma-ring \mathfrak{M}(\mu). This extended set function is called a measure. The special case \mu = m is called the Lebesgue measure on \mathbb{R}^p.

{Remarks}

• If A is open, then A\in\mathfrak{M}(\mu). By taking complements, it follows that every closed set is in \mathfrak{M}(\mu).
  
• If A\in\mathfrak{M} and \varepsilon>0, there exists closed set F and open set G such that F\subset A\subset G and \mu(G-A)<\varepsilon, \mu(A-F)<\varepsilon.
  
• We say that E is a Borel set if E can be obtained by a countable number of operations, starting from open sets, each operation consisting in taking unions, intersections, or complements. The collection \mathcal{B} of all Borel sets in \mathbb{R}^p is a \sigma-ring; in fact, it is the smallest \sigma-ring contains all open sets. By the first remark, E\in\mathfrak{M}(\mu) if E\in\mathcal{B}.
  
• If A\in\in\mathfrak{M}(\mu), there exist Borel sets F and G such that F\subset A\subset G, and \mu(G-A)=\mu(A-F)=0.

  We see that every A\in\in\mathfrak{M}(\mu) is the union of a Borel set and a set of measure zero.

  The Borel sets are \mu-measurable for every \mu. But the sets of measure zero may be different for different \mu's.
  

• For every \mu, the sets of measure zero form a \sigma-ring.
  
• In case of the Lebesgue measure, every countable set has measure zero. But there are uncountable (in fact, perfect) sets of measure zero. The Cantor set may be taken as an example.

For instance, let X be the set of all positive integers, \mathfrak{M} the collection of all subsets of X, and \mu(E) the number of elements of E.

In the following sections we shall always deal with measurable spaces. It should be emphasized that the integration theory which we shall soon discuss would not become simpler in any respect if we sacrificed the generality we have now attained and restricted ourselves to Lebesgue measure, say, on an interval of the real line. In fact, the essential features of the theory are brought out with much greater clarity in the more general situation, where it is seen that everything depends only on the countable additivity of \mu on a \sigma-ring.
4. Measurable functions

Summing up, we may say that all ordinary operations of analysis, including limit operations, when applied to measurable functions, lead to measurable functions; in other words, all functions that are ordinarily met with are measurable.

That this is, however, only a rough statement is shown by the following example (based on Lebesgue measure, on the real line): If h(x) = f(g(x)), where f is measurable and g is continuous, then h is not necessarily measurable.

One may have noticed that measure has not been mentioned in our discussion of measurable functions. In fact, the class of measurable functions on X depends only on the \sigma-ring \mathfrak{M}. For instance, we may speak of Borel-measurable functions on \mathbb{R}^p, that is, of function f for which \{x\mid f(x)>a\} is always a Borel set, without reference to any particular measure.
5. Simple functions

Suppose the range of s consists of the distinct numbers c_1, \dots , c_n. Let E_i=\{x\mid s(x)=c_i\}\, (i=1,\dots,n). Then s=\sum c_iK_{E_i}, that is, every simple function is a finite linear combination of characteristic functions. It is clear that s is measurable if and only if the sets E_1, \dots , E_n are measurable.

It is of interest that every function can be approximated by simple functions:

We shall define integration on a measurable space X, in which \mathfrak{M} is the \sigma-ring of measurable sets, and \mu is the measure. The one who wishes to visualize a more concrete situation may think of X as the real line, or an interval, and of \mu as the Lebesgue measure m.

It is easy to verify that \int_E s\, d\mu=I_E(s) for every nonnegative simple measurable functions s.

This terminology may be a little confusing: If \int_Ef\, d\mu is +\infty or -\infty, then the integral of f over E is defined, although f is not integrable in the above sense of the word; f is integrable on E only if its integral over E is finite.

We shall be mainly interested in integrable functions, although in some cases it is desirable to deal with the more general situation.

The following properties are evident:

• If f is measurable and bounded on E, and if \mu(E)<+\infty, then f\in\mathcal{L}(\mu) on E.
  
• If a\leqslant f\leqslant b for x\in E, and \mu(E)<+\infty, then a \mu(E) \leq \int_{E} f\, d \mu \leq b \mu(E).
  
• If f and g \in\mathcal{L}(\mu) on E, and if f(x)\leqslant g(x) for x\in E, then \int_E f\, d\mu\leqslant \int_Eg\, d\mu.
  
• If f\in\mathcal{L}(\mu) on E, then cf\in\mathcal{L}(\mu) on E, for every finite constant c, and \int_{E} c f\, d \mu=c \int_{E} f\, d \mu.
  
• If \mu(E)=0, and f is measurable, then \int_Ef\, d\mu=0.
  
• If f\in\mathcal{L}(\mu) on E, A\in\mathfrak{M}, and A\subset E, then f\in\mathcal{L}(\mu) on A.

The preceding corollary shows that sets of measure zero are negligible in integration.

If f\in\mathcal{L}(\mu) on E, it is clear that f(x) must be finite almost everywhere on E. In most cases we therefore do not lose any generality if we assume the given functions to be finite-valued from the outset.

Since the integrability of f implies that of |f|, the Lebesgue integral is often called an absolutely convergent integral. It is of course possible to define nonabsolutely convergent integrals, and in the treatment of some problems it is essential to do so. But these integrals lack some of the most useful properties of the Lebesgue integral and play a somewhat less important role in analysis.

A uniformly bounded convergent sequence is often said to be boundedly convergent.
7. Comparison with the Riemann integral Our next theorem will show that every function which is Riemann-integrable on an interval is also Lebesgue-integrable, and that Riemann-integrable functions are subject to rather stringent continuity conditions. Quite apart from the fact that the Lebesgue theory therefore enables us to integrate a much larger class of functions, its greatest advantage lies perhaps in the ease with which many limit operations can be handled; from this point of view, Lebesgue's convergence theorems may well be regarded as the core of the Lebesgue theory.

One of the difficulties which is encountered in the Riemann theory is that limits of Riemann-integrable functions (or even continuous functions) may fail to be Riemann-integrable. This difficulty is now almost eliminated, since limits of measurable functions are always measurable.

Let the measure space X be the interval [a, b] of the real line, with \mu = m (the Lebesgue measure), and \mathfrak{M} the family of Lebesgue-measurable subsets of [a, b]. Instead of \int_Xf\, dm it is customary to use the familiar notation \int_{a}^{b}f\, dx for the Lebesgue integral of f over [a, b]. To distinguish Riemann integrals from Lebesgue integrals, we shall now denote the former by \mathcal{R}\int_{a}^{b}f\, dx.

The familiar connection between integration and differentiation is to a large degree carried over into the Lebesgue theory. If f\in\mathcal{L} on [a, b], and F(x)=\int_{a}^{x}f\, dt\, (a\leqslant x\leqslant b), then F^\prime(x)=f(x) almost everywhere on [a,b].

Conversely, if F is differentiable at every point of [a, b] ("almost everywhere" is not good enough here!) and if F^\prime\in\mathcal{L} on [a, b], then F(x)-F(a)=\int_{a}^{x}F^\prime(t)\, (a\leqslant x\leqslant b).
8. Integration of complex functions Suppose f is a complex-valued function defined on a measure space X, and f = u + iv, where u and v are real. We say that f is measurable if and only if both u and v are measurable.

It is easy to verify that sums and products of complex measurable functions are again measurable. Since |f|=\sqrt{u^2+v^2}, |f| is measurable for every complex measurable f.

Suppose \mu is a measure on X, E is a measurable subset of X, and f is a complex function on X. We say that f\in\mathcal{L}(E) on E provided that f is measurable and \int_E|f|\, d\mu<+\infty, and we define
\[\int_Ef\, d\mu=\int_Eu\, d\mu+i\int_Ev\, d\mu\]
if \int_E|f|\, d\mu<+\infty holds. Since |u|\leqslant|f|, |v|\leqslant|f|, and |f|\leqslant|u|+|v|, it is clear that \int_E|f|\, d\mu<+\infty holds if and only if u\in\mathcal{L}(\mu) and v\in\mathcal{L}(\mu) on E.

Many theorems stated before can now be extended to Lebesgue integrals of complex functions. The proofs are quite straightforward. the only one that offers anything of interest is:

If f\in\mathcal{L}(\mu) on E, there is a complex number c, |c|=1, such that c\int_E f\, d\mu\geqslant0. Put g=cf=u+iv, u and v real. Then
\[\left|\int_Ef\, d\mu\right|=c\int_Ef\, d\mu=\int_Eg\, d\mu=\int_Eu\, d\mu\leqslant\int_E|f|\, d\mu.\]
9. Functions of class \mathcal{L}^2 As an application of the Lebesgue theory, we shall now extend the Parseval theorem (which we proved only for Riemann-integrable functions in Chapter 8) and prove the Riesz-Fischer theorem for orthonormal sets of functions.

This is the Schwarz inequality, which we have already encountered for series and for Riemann integrals. It follows from the inequality
\[0 \leqslant \int_{X}(|f|+\lambda|g|)^{2}\, d \mu=|f|^{2}+2 \lambda \int_{X}|f g|\, d \mu+\lambda^{2}|g|^{2},\]
which holds for every real \lambda.

Schwarz inequality shows that
\[|f+g|^{2} =\int|f|^{2}+\int f \bar{g}+\int f g+\int|g|^{2}\leqslant|f|^{2}+2|f||g|+|g|^{2} =(|f|+|g|)^{2}\]

If we define the distance between two functions f and g in \mathcal{L}^2(\mu) to be |f-g|, we see that the conditions of definition of metric space are satisfied, except for the fact that |f-g| = 0 does not imply that f(x) = g(x) for all x, but only for almost all x. Thus, if we identify functions which differ only on a set of measure zero, \mathcal{L}^2(\mu) is a metric space.

We now consider \mathcal{L}^2 on an interval of the real line, with respect to Lebesgue measure.

\textbf{Proof. }We shall say f is approximated in \mathcal{L}^2 by a sequence \{g_n\} if |f-g_n|\to0 as n\to\infty.

Let A be a closed subset of [a,b], and K_A its characteristic function. Put t(x)=\inf|x-y|\, (y\in A) and g_n(x)=1/(1+nt(x))\, n=1,2,\dots. Then g_n is continuous on [a,b], g_n(x)=1 on A, and g_n(x)\to0 on B, where B=[a,b]-A. Hence by dominated convergence theorem, |g_n-K_A|=\{\int_Bg_n^2\, dx\}^{1/2}\to0. Thus characteristic functions of closed sets can be approximated in \mathcal{L}^2 by continuous functions.

The same is true for the characteristic function of any measurable set, and hence also for simple measurable functions.

If f\geqslant0 and f\in\mathcal{L}^2, let \{s_n\} be a monotonically increasing sequence of simple nonnegative measurable functions such that s_n(x)\to f(x). Since |f-s_n|\leqslant f^2, dominated convergence theorem shows that |f-s_n|\to0. The general case follows.

The definition of a trigonometric Fourier series is extended in the same way to \mathcal{L}^2 (or even to \mathcal{L}) on [-n, n]. Theorems 8.5.1 and 8.5.2 (the Bessel inequality) hold for any f\in\mathcal{L}^2(\mu). The proofs are the same, word for word.

Thus if two functions in \mathcal{L}^2 have the same Fourier series, they differ at most on a set of measure zero.

We have previously deduced the completeness of the trigonometric system from the Parseval equation. Conversely, the Parseval
equation holds for every complete orthonormal set:

Combining the Riesz-Fischer theorem and this theorem, we arrive at the very interesting conclusion that every complete orthonormal set induces a 1-1 correspondence between the functions f\in \mathcal{L}^2(\mu) (identifying those which are equal almost everywhere) on the one hand and the sequences \{c_n\} for which \sum|c_n|^2 converges, on the other. The representation f\sim \sum c_n\phi_n together with the Parseval equation, shows that \mathcal{L}^2(\mu) may be regarded as an infinite-dimensional Euclidean space (the so-called "Hilbert space"), in which the point f has coordinates c_n, and the functions \phi_n are the coordinate vectors.