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245A, Notes 4: Modes of convergence

2 October, 2010 in 245A - Real analysis, math.CA | Tags: , , , , , | by | 50 comments

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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245B, notes 0: A quick review of measure and integration theory

1 January, 2009 in 245B - Real analysis, math.CA | Tags: , , , , | by | 46 comments

In these notes we quickly review the basics of abstract measure theory and integration theory, which was covered in the previous course but will of course be relied upon in the current course.  This is only a brief summary of the material; of course, one should consult a real analysis text for the full details of the theory.

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