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245A, Notes 2: The Lebesgue integral

19 September, 2010 in 245A - Real analysis, math.CA | Tags: , , , , | by | 109 comments

In the previous notes, we defined the Lebesgue measure {m(E)} of a Lebesgue measurable set {E \subset {\bf R}^d}, and set out the basic properties of this measure. In this set of notes, we use Lebesgue measure to define the Lebesgue integral

\displaystyle \int_{{\bf R}^d} f(x)\ dx

of functions {f: {\bf R}^d \rightarrow {\bf C} \cup \{\infty\}}. 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 {[0,+\infty]}), 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

\displaystyle \sum_{n=1}^\infty c_n

of a sequence of numbers {c_n}, which can be viewed as a discrete analogue of the Lebesgue integral. 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 {c_n} lie in the extended non-negative real axis {[0,+\infty]}. In this case, the infinite sum can be defined as the limit of the partial sums

\displaystyle \sum_{n=1}^\infty c_n = \lim_{N \rightarrow \infty} \sum_{n=1}^N c_n \ \ \ \ \ (1)

 

or equivalently as a supremum of arbitrary finite partial sums:

\displaystyle \sum_{n=1}^\infty c_n = \sup_{A \subset {\bf N}, A \hbox{ finite}} \sum_{n \in A} c_n. \ \ \ \ \ (2)

 

The unsigned infinite sum {\sum_{n=1}^\infty c_n} always exists, but its value may be infinite, even when each term is individually finite (consider e.g. {\sum_{n=1}^\infty 1}).

The second notion of a summation is the absolutely summable infinite sum, in which the {c_n} lie in the complex plane {{\bf C}} and obey the absolute summability condition

\displaystyle \sum_{n=1}^\infty |c_n| < \infty,

where the left-hand side is of course an unsigned infinite sum. When this occurs, one can show that the partial sums {\sum_{n=1}^N c_n} 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 {{\bf C}} rather than {[0,+\infty]}. 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

\displaystyle \sum_{n=1}^\infty c_n = (\sum_{n=1}^\infty \hbox{Re}(c_n)) + i (\sum_{n=1}^\infty \hbox{Im}(c_n))

for complex absolutely summable {c_n}, and

\displaystyle \sum_{n=1}^\infty c_n = \sum_{n=1}^\infty c_n^+ - \sum_{n=1}^\infty c_n^-

for real absolutely summable {c_n}, where {c_n^+ := \max(c_n,0)} and {c_n^- := \max(-c_n,0)} are the (magnitudes of the) positive and negative parts of {c_n}.

In an analogous spirit, we will first define an unsigned Lebesgue integral {\int_{{\bf R}^d} f(x)\ dx} of (measurable) unsigned functions {f: {\bf R}^d \rightarrow [0,+\infty]}, and then use that to define the absolutely convergent Lebesgue integral {\int_{{\bf R}^d} f(x)\ dx} of absolutely integrable functions {f: {\bf R}^d \rightarrow {\bf C} \cup \{\infty\}}. (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 {\int_a^b f(x)\ dx} of a Riemann integrable function {f: [a,b] \rightarrow {\bf R}}. Recall from the prologue that this integral is equal to the lower Darboux integral

\displaystyle \int_a^b f(x) = \underline{\int_a^b} f(x)\ dx := \sup_{g \leq f; g \hbox{ piecewise constant}} \hbox{p.c.} \int_a^b g(x)\ dx.

(It is also equal to the upper Darboux integral; but much as the theory of Lebesgue measure is easiest to define by relying solely on outer measure and not on inner measure, the theory of the unsigned Lebesgue integral is easiest to define by relying solely on lower integrals rather than upper ones; the upper integral is somewhat problematic when dealing with “improper” integrals of functions that are unbounded or are supported on sets of infinite measure.) Compare this formula also with (2). The integral {\hbox{p.c.} \int_a^b g(x)\ dx} is a piecewise constant integral, formed by breaking up the piecewise constant functions {g, h} 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 {\underline{\int_{{\bf R}^d}} f(x)\ dx} of any unsigned function {f: {\bf R}^d \rightarrow [0,+\infty]}, 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. As we shall see, it obeys all the basic properties one expects of an integral, such as monotonicity and additivity; in subsequent notes we will also see that it behaves quite well with respect to limits, as we shall see by establishing the two basic convergence theorems of the unsigned Lebesgue integral, namely Fatou’s lemma and the monotone convergence theorem.

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 {{\bf R}^d}) 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 {L^1} 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 {L^p} spaces, but the convergence theorems mentioned above already hint at this completeness. A related fact, known as Egorov’s theorem, asserts that a pointwise converging sequence of functions can be approximated as a (locally) uniformly converging sequence of functions. The facts listed here manifestations of Littlewood’s three principles of real analysis, which capture much of the essence of the Lebesgue theory.

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