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If one has a sequence of real numbers
, it is unambiguous what it means for that sequence to converge to a limit
: it means that for every
, there exists an
such that
for all
. Similarly for a sequence
of complex numbers
converging to a limit
.
More generally, if one has a sequence of
-dimensional vectors
in a real vector space
or complex vector space
, it is also unambiguous what it means for that sequence to converge to a limit
or
; it means that for every
, there exists an
such that
for all
. Here, the norm
of a vector
can be chosen to be the Euclidean norm
, the supremum norm
, 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
or
.
If however one has a sequence of functions
or
on a common domain
, and a putative limit
or
, there can now be many different ways in which the sequence
may or may not converge to the limit
. (One could also consider convergence of functions
on different domains
, but we will not discuss this issue at all here.) This is contrast with the situation with scalars
or
(which corresponds to the case when
is a single point) or vectors
(which corresponds to the case when
is a finite set such as
). Once
becomes infinite, the functions
acquire an infinite number of degrees of freedom, and this allows them to approach
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:
- We say that
converges to
pointwise if, for every
,
converges to
. In other words, for every
and
, there exists
(that depends on both
and
) such that
whenever
.
- We say that
converges to
uniformly if, for every
, there exists
such that for every
,
for every
. The difference between uniform convergence and pointwise convergence is that with the former, the time
at which
must be permanently
-close to
is not permitted to depend on
, but must instead be chosen uniformly in
.
Uniform convergence implies pointwise convergence, but not conversely. A typical example: the functions defined by
converge pointwise to the zero function
, 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 is equipped with the structure of a measure space
, and the functions
(and their limit
) are measurable with respect to this space. In this context, we have some additional modes of convergence:
- We say that
converges to
pointwise almost everywhere if, for (
-)almost everywhere
,
converges to
.
- We say that
converges to
uniformly almost everywhere, essentially uniformly, or in
norm if, for every
, there exists
such that for every
,
for
-almost every
.
- We say that
converges to
almost uniformly if, for every
, there exists an exceptional set
of measure
such that
converges uniformly to
on the complement of
.
- We say that
converges to
in
norm if the quantity
converges to
as
.
- We say that
converges to
in measure if, for every
, the measures
converge to zero as
.
Observe that each of these five modes of convergence is unaffected if one modifies or
on a set of measure zero. In contrast, the pointwise and uniform modes of convergence can be affected if one modifies
or
even on a single point.
Remark 1 In the context of probability theory, in which
and
are interpreted as random variables, convergence in
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
be a measure space, let
be sequences of measurable functions, and let
be measurable functions.
- Show that
converges to
along one of the above seven modes of convergence if and only if
converges to
along the same mode.
- If
converges to
along one of the above seven modes of convergence, and
converges to
along the same mode, show that
converges to
along the same mode, and that
converges to
along the same mode for any
.
- (Squeeze test) If
converges to
along one of the above seven modes, and
pointwise for each
, show that
converges to
along the same mode.
We have some easy implications between modes:
Exercise 2 (Easy implications) Let
be a measure space, and let
and
be measurable functions.
- If
converges to
uniformly, then
converges to
pointwise.
- If
converges to
uniformly, then
converges to
in
norm. Conversely, if
converges to
in
norm, then
converges to
uniformly outside of a null set (i.e. there exists a null set
such that the restriction
of
to the complement of
converges to the restriction
of
).
- If
converges to
in
norm, then
converges to
almost uniformly.
- If
converges to
almost uniformly, then
converges to
pointwise almost everywhere.
- If
converges to
pointwise, then
converges to
pointwise almost everywhere.
- If
converges to
in
norm, then
converges to
in measure.
- If
converges to
almost uniformly, then
converges to
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 is the real line
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
. Then
converges to zero pointwise (and thus, pointwise almost everywhere), but not uniformly, in
norm, almost uniformly, in
norm, or in measure.
Example 2 (Escape to width infinity) Let
. Then
converges to zero uniformly (and thus, pointwise, pointwise almost everywhere, in
norm, almost uniformly, and in measure), but not in
norm.
Example 3 (Escape to vertical infinity) Let
. Then
converges to zero pointwise (and thus, pointwise almost everywhere) and almost uniformly (and hence in measure), but not uniformly, in
norm, or in
norm.
Example 4 (Typewriter sequence) Let
be defined by the formula
whenever
and
. This is a sequence of indicator functions of intervals of decreasing length, marching across the unit interval
over and over again. Then
converges to zero in measure and in
norm, but not pointwise almost everywhere (and hence also not pointwise, not almost uniformly, nor in
norm, nor uniformly).
Remark 2 The
norm
of a measurable function
is defined to the infimum of all the quantities
that are essential upper bounds for
in the sense that
for almost every
. Then
converges to
in
norm if and only if
as
. The
and
norms are part of the larger family of
norms, which we will study in more detail in 245B.
One particular advantage of convergence is that, in the case when the
are absolutely integrable, it implies convergence of the integrals,
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 or a uniform integrability hypothesis, then one can obtain some additional implications between the different modes.
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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