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If one has a sequence {x_1, x_2, x_3, \ldots \in {\bf R}} of real numbers {x_n}, it is unambiguous what it means for that sequence to converge to a limit {x \in {\bf R}}: it means that for every {\epsilon > 0}, there exists an {N} such that {|x_n-x| \leq \epsilon} for all {n > N}. Similarly for a sequence {z_1, z_2, z_3, \ldots \in {\bf C}} of complex numbers {z_n} converging to a limit {z \in {\bf C}}.

More generally, if one has a sequence {v_1, v_2, v_3, \ldots} of {d}-dimensional vectors {v_n} in a real vector space {{\bf R}^d} or complex vector space {{\bf C}^d}, it is also unambiguous what it means for that sequence to converge to a limit {v \in {\bf R}^d} or {v \in {\bf C}^d}; it means that for every {\epsilon > 0}, there exists an {N} such that {\|v_n-v\| \leq \epsilon} for all {n \geq N}. Here, the norm {\|v\|} of a vector {v = (v^{(1)},\ldots,v^{(d)})} can be chosen to be the Euclidean norm {\|v\|_2 := (\sum_{j=1}^d (v^{(j)})^2)^{1/2}}, the supremum norm {\|v\|_\infty := \sup_{1 \leq j \leq d} |v^{(j)}|}, or any other number of norms, but for the purposes of convergence, these norms are all equivalent; a sequence of vectors converges in the Euclidean norm if and only if it converges in the supremum norm, and similarly for any other two norms on the finite-dimensional space {{\bf R}^d} or {{\bf C}^d}.

If however one has a sequence {f_1, f_2, f_3, \ldots} of functions {f_n: X \rightarrow {\bf R}} or {f_n: X \rightarrow {\bf C}} on a common domain {X}, and a putative limit {f: X \rightarrow {\bf R}} or {f: X \rightarrow {\bf C}}, there can now be many different ways in which the sequence {f_n} may or may not converge to the limit {f}. (One could also consider convergence of functions {f_n: X_n \rightarrow {\bf C}} on different domains {X_n}, but we will not discuss this issue at all here.) This is contrast with the situation with scalars {x_n} or {z_n} (which corresponds to the case when {X} is a single point) or vectors {v_n} (which corresponds to the case when {X} is a finite set such as {\{1,\ldots,d\}}). Once {X} becomes infinite, the functions {f_n} acquire an infinite number of degrees of freedom, and this allows them to approach {f} in any number of inequivalent ways.

What different types of convergence are there? As an undergraduate, one learns of the following two basic modes of convergence:

  1. We say that {f_n} converges to {f} pointwise if, for every {x \in X}, {f_n(x)} converges to {f(x)}. In other words, for every {\epsilon > 0} and {x \in X}, there exists {N} (that depends on both {\epsilon} and {x}) such that {|f_n(x)-f(x)| \leq \epsilon} whenever {n \geq N}.
  2. We say that {f_n} converges to {f} uniformly if, for every {\epsilon > 0}, there exists {N} such that for every {n \geq N}, {|f_n(x) - f(x)| \leq \epsilon} for every {x \in X}. The difference between uniform convergence and pointwise convergence is that with the former, the time {N} at which {f_n(x)} must be permanently {\epsilon}-close to {f(x)} is not permitted to depend on {x}, but must instead be chosen uniformly in {x}.

Uniform convergence implies pointwise convergence, but not conversely. A typical example: the functions {f_n: {\bf R} \rightarrow {\bf R}} defined by {f_n(x) := x/n} converge pointwise to the zero function {f(x) := 0}, but not uniformly.

However, pointwise and uniform convergence are only two of dozens of many other modes of convergence that are of importance in analysis. We will not attempt to exhaustively enumerate these modes here (but see this Wikipedia page, and see also these 245B notes on strong and weak convergence). We will, however, discuss some of the modes of convergence that arise from measure theory, when the domain {X} is equipped with the structure of a measure space {(X, {\mathcal B}, \mu)}, and the functions {f_n} (and their limit {f}) are measurable with respect to this space. In this context, we have some additional modes of convergence:

  1. We say that {f_n} converges to {f} pointwise almost everywhere if, for ({\mu}-)almost everywhere {x \in X}, {f_n(x)} converges to {f(x)}.
  2. We say that {f_n} converges to {f} uniformly almost everywhere, essentially uniformly, or in {L^\infty} norm if, for every {\epsilon > 0}, there exists {N} such that for every {n \geq N}, {|f_n(x) - f(x)| \leq \epsilon} for {\mu}-almost every {x \in X}.
  3. We say that {f_n} converges to {f} almost uniformly if, for every {\epsilon > 0}, there exists an exceptional set {E \in {\mathcal B}} of measure {\mu(E) \leq \epsilon} such that {f_n} converges uniformly to {f} on the complement of {E}.
  4. We say that {f_n} converges to {f} in {L^1} norm if the quantity {\|f_n-f\|_{L^1(\mu)} = \int_X |f_n(x)-f(x)|\ d\mu} converges to {0} as {n \rightarrow \infty}.
  5. We say that {f_n} converges to {f} in measure if, for every {\epsilon > 0}, the measures {\mu( \{ x \in X: |f_n(x) - f(x)| \geq \epsilon \} )} converge to zero as {n \rightarrow \infty}.

Observe that each of these five modes of convergence is unaffected if one modifies {f_n} or {f} on a set of measure zero. In contrast, the pointwise and uniform modes of convergence can be affected if one modifies {f_n} or {f} even on a single point.

Remark 1 In the context of probability theory, in which {f_n} and {f} are interpreted as random variables, convergence in {L^1} norm is often referred to as convergence in mean, pointwise convergence almost everywhere is often referred to as almost sure convergence, and convergence in measure is often referred to as convergence in probability.

Exercise 1 (Linearity of convergence) Let {(X, {\mathcal B}, \mu)} be a measure space, let {f_n, g_n: X \rightarrow {\bf C}} be sequences of measurable functions, and let {f, g: X \rightarrow {\bf C}} be measurable functions.

  1. Show that {f_n} converges to {f} along one of the above seven modes of convergence if and only if {|f_n-f|} converges to {0} along the same mode.
  2. If {f_n} converges to {f} along one of the above seven modes of convergence, and {g_n} converges to {g} along the same mode, show that {f_n+g_n} converges to {f+g} along the same mode, and that {cf_n} converges to {cf} along the same mode for any {c \in {\bf C}}.
  3. (Squeeze test) If {f_n} converges to {0} along one of the above seven modes, and {|g_n| \leq f_n} pointwise for each {n}, show that {g_n} converges to {0} along the same mode.

We have some easy implications between modes:

Exercise 2 (Easy implications) Let {(X, {\mathcal B}, \mu)} be a measure space, and let {f_n: X \rightarrow {\bf C}} and {f: X \rightarrow {\bf C}} be measurable functions.

  1. If {f_n} converges to {f} uniformly, then {f_n} converges to {f} pointwise.
  2. If {f_n} converges to {f} uniformly, then {f_n} converges to {f} in {L^\infty} norm. Conversely, if {f_n} converges to {f} in {L^\infty} norm, then {f_n} converges to {f} uniformly outside of a null set (i.e. there exists a null set {E} such that the restriction {f_n\downharpoonright_{X \backslash E}} of {f_n} to the complement of {E} converges to the restriction {f\downharpoonright_{X \backslash E}} of {f}).
  3. If {f_n} converges to {f} in {L^\infty} norm, then {f_n} converges to {f} almost uniformly.
  4. If {f_n} converges to {f} almost uniformly, then {f_n} converges to {f} pointwise almost everywhere.
  5. If {f_n} converges to {f} pointwise, then {f_n} converges to {f} pointwise almost everywhere.
  6. If {f_n} converges to {f} in {L^1} norm, then {f_n} converges to {f} in measure.
  7. If {f_n} converges to {f} almost uniformly, then {f_n} converges to {f} in measure.

The reader is encouraged to draw a diagram that summarises the logical implications between the seven modes of convergence that the above exercise describes.

We give four key examples that distinguish between these modes, in the case when {X} is the real line {{\bf R}} with Lebesgue measure. The first three of these examples already were introduced in the previous set of notes.

Example 1 (Escape to horizontal infinity) Let {f_n := 1_{[n,n+1]}}. Then {f_n} converges to zero pointwise (and thus, pointwise almost everywhere), but not uniformly, in {L^\infty} norm, almost uniformly, in {L^1} norm, or in measure.

Example 2 (Escape to width infinity) Let {f_n := \frac{1}{n} 1_{[0,n]}}. Then {f_n} converges to zero uniformly (and thus, pointwise, pointwise almost everywhere, in {L^\infty} norm, almost uniformly, and in measure), but not in {L^1} norm.

Example 3 (Escape to vertical infinity) Let {f_n := n 1_{[\frac{1}{n}, \frac{2}{n}]}}. Then {f_n} converges to zero pointwise (and thus, pointwise almost everywhere) and almost uniformly (and hence in measure), but not uniformly, in {L^\infty} norm, or in {L^1} norm.

Example 4 (Typewriter sequence) Let {f_n} be defined by the formula

\displaystyle  f_n := 1_{[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}]}

whenever {k \geq 0} and {2^k \leq n < 2^{k+1}}. This is a sequence of indicator functions of intervals of decreasing length, marching across the unit interval {[0,1]} over and over again. Then {f_n} converges to zero in measure and in {L^1} norm, but not pointwise almost everywhere (and hence also not pointwise, not almost uniformly, nor in {L^\infty} norm, nor uniformly).

Remark 2 The {L^\infty} norm {\|f\|_{L^\infty(\mu)}} of a measurable function {f: X \rightarrow {\bf C}} is defined to the infimum of all the quantities {M \in [0,+\infty]} that are essential upper bounds for {f} in the sense that {|f(x)| \leq M} for almost every {x}. Then {f_n} converges to {f} in {L^\infty} norm if and only if {\|f_n-f\|_{L^\infty(\mu)} \rightarrow 0} as {n \rightarrow \infty}. The {L^\infty} and {L^1} norms are part of the larger family of {L^p} norms, which we will study in more detail in 245B.

One particular advantage of {L^1} convergence is that, in the case when the {f_n} are absolutely integrable, it implies convergence of the integrals,

\displaystyle  \int_X f_n\ d\mu \rightarrow \int_X f\ d\mu,

as one sees from the triangle inequality. Unfortunately, none of the other modes of convergence automatically imply this convergence of the integral, as the above examples show.

The purpose of these notes is to compare these modes of convergence with each other. Unfortunately, the relationship between these modes is not particularly simple; unlike the situation with pointwise and uniform convergence, one cannot simply rank these modes in a linear order from strongest to weakest. This is ultimately because the different modes react in different ways to the three “escape to infinity” scenarios described above, as well as to the “typewriter” behaviour when a single set is “overwritten” many times. On the other hand, if one imposes some additional assumptions to shut down one or more of these escape to infinity scenarios, such as a finite measure hypothesis {\mu(X) < \infty} or a uniform integrability hypothesis, then one can obtain some additional implications between the different modes.

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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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