If one has a sequence x1, x2, x3, . . . ∈ R of real numbers xn, it is
unambiguous what it means for that sequence to converge to a limit
x∈ R: it means that for every ε > 0, there exists an N such that |xn−x| ≤εfor alln > N. Similarly for a sequencez1, z2, z3, . . .∈C
of complex numberszn converging to a limitz∈C.
More generally, if one has a sequencev1, v2, v3, . . .ofd-dimensional
vectorsvn in a real vector space Rd or complex vector spaceCd, it
is also unambiguous what it means for that sequence to converge to a limit v ∈ Rd or v ∈ Cd; it means that for every ε > 0,
there exists an N such that kvn −vk ≤ ε for all n ≥ N. Here,
the norm kvk of a vector v = (v(1), . . . , v(d)) can be chosen to be
the Euclidean norm kvk2 := (P
d j=1(v
(j))2)1/2, the supremum norm
kvk∞ := sup1≤j≤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 spaceRd orCd.
If however one has a sequencef1, f2, f3, . . .of functionsfn: X→
R or fn: X → C on a common domain X, and a putative limit
f: X →Ror f:X →C, there can now be many different ways in which the sequencefn may or may not converge to the limitf. (One
could also consider convergence of functionsfn:Xn→Con different
domains Xn, but we will not discuss this issue at all here.) This is
contrast with the situation with scalarsxn orzn (which corresponds
to the case whenX is a single point) or vectorsvn(which corresponds
to the case when X is a finite set such as {1, . . . , d}). Once X be- comes infinite, the functionsfn 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 undergrad- uate, one learns of the following two basicmodes of convergence:
(i) We say thatfn converges tof pointwiseif, for everyx∈X,
fn(x) converges to f(x). In other words, for every ε > 0
andx∈X, there existsN (that depends onboth εand x) such that|fn(x)−f(x)| ≤εwhenevern≥N.
(ii) We say thatfn converges tof uniformlyif, for everyε >0,
there existsN such that for everyn≥N,|fn(x)−f(x)| ≤ε
for every x∈ X. The difference between uniform conver- gence and pointwise convergence is that with the former, the timeN at whichfn(x) must be permanentlyε-close to
f(x) is not permitted to depend onx, but must instead be chosen uniformly inx.
Uniform convergence implies pointwise convergence, but not con- versely. A typical example: the functions fn: R → R defined by
fn(x) :=x/nconverge pointwise to the zero functionf(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 §1.9 of An epsilon of room, Vol. I). 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,B, µ), and the functionsfn (and their limitf)
are measurable with respect to this space. In this context, we have some additional modes of convergence:
(i) We say thatfn converges tof pointwise almost everywhere
if, for (µ-)almost everywhere x ∈ X, fn(x) converges to
f(x).
(ii) We say thatfnconverges tof uniformly almost everywhere, essentially uniformly, or in L∞ norm if, for every ε > 0, there existsN such that for everyn≥N,|fn(x)−f(x)| ≤ε
forµ-almost everyx∈X.
(iii) We say thatfnconverges tof almost uniformlyif, for every
ε > 0, there exists an exceptional set E ∈ B of measure
µ(E) ≤ ε such that fn converges uniformly to f on the
complement ofE.
(iv) We say thatfn converges to f in L1 norm if the quantity
kfn−fkL1(µ)=R
X|fn(x)−f(x)|dµconverges to 0 asn→
∞.
(v) We say thatfnconverges tof in measureif, for everyε >0,
the measuresµ({x∈ X :|fn(x)−f(x)| ≥ ε}) converge to
zero asn→ ∞.
Observe that each of these five modes of convergence is unaffected if one modifies fn or f on a set of measure zero. In contrast, the
pointwise and uniform modes of convergence can be affected if one modifiesfn or f even on a single point. The L1 and L∞ modes of
converges are special cases of theLp mode of convergence, which is discussed in§1.3 ofAn epsilon of room, Vol. I.
Remark 1.5.1. In the context of probability theory (see Section 2.3), in whichfn and f are interpreted as random variables, convergence
inL1norm is often referred to asconvergence in mean, pointwise con-
vergence almost everywhere is often referred to asalmost sure conver- gence, and convergence in measure is often referred to asconvergence in probability.
Exercise 1.5.1 (Linearity of convergence). Let (X,B, µ) be a mea- sure space, letfn, gn:X →C be sequences of measurable functions,
and letf, g:X→C be measurable functions.
(i) Show thatfn converges to f along one of the above seven
modes of convergence if and only if|fn−f| converges to 0
along the same mode.
(ii) If fn converges tof along one of the above seven modes of
convergence, and gn converges to g along the same mode,
show thatfn+gn converges tof+galong the same mode,
and thatcfn converges tocf along the same mode for any
c∈C.
(iii) (Squeeze test) If fn converges to 0 along one of the above
seven modes, and|gn| ≤fn pointwise for eachn, show that
gn converges to 0 along the same mode.
We have some easy implications between modes:
Exercise 1.5.2(Easy implications).Let (X,B, µ) be a measure space, and letfn:X→Candf:X→C be measurable functions.
(i) Iffnconverges tof uniformly, thenfnconverges tof point-
wise.
(ii) Iffnconverges tof uniformly, thenfnconverges tof inL∞
norm. Conversely, iffn converges to f in L∞ norm, then
fn converges to f uniformly outside of a null set (i.e. there
exists a null set E such that the restriction fn X\E of fn
to the complement ofE converges to the restrictionf X\E
off).
(iii) If fn converges to f in L∞ norm, then fn converges to f
almost uniformly.
(iv) Iffn converges tof almost uniformly, thenfn converges to
f pointwise almost everywhere.
(v) Iffnconverges tof pointwise, thenfnconverges tof point-
wise almost everywhere.
(vi) Iffn converges tof in L1 norm, thenfn converges tof in
(vii) If fn converges tof almost uniformly, thenfn 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 R with Lebesgue measure. The first three of these examples already were introduced in Section 1.4, but the fourth is new, and also important.
Example 1.5.2 (Escape to horizontal infinity). Letfn := 1[n,n+1].
Thenfn converges to zero pointwise (and thus, pointwise almost ev-
erywhere), but not uniformly, in L∞ norm, almost uniformly, in L1
norm, or in measure.
Example 1.5.3 (Escape to width infinity). Letfn:= n11[0,n]. Then
fnconverges to zero uniformly (and thus, pointwise, pointwise almost
everywhere, inL∞norm, almost uniformly, and in measure), but not inL1 norm.
Example 1.5.4 (Escape to vertical infinity). Let fn := n1[1
n,2n]. Thenfn converges to zero pointwise (and thus, pointwise almost ev-
erywhere) and almost uniformly (and hence in measure), but not uniformly, inL∞ norm, or inL1 norm.
Example 1.5.5 (Typewriter sequence). Let fn be defined by the
formula
fn:= 1[n−2k
2k , n−2k+1
2k ]
wheneverk≥0 and 2k ≤n <2k+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 fn converges to zero in
measure and inL1norm, but not pointwise almost everywhere (and
hence also not pointwise, not almost uniformly, nor inL∞ norm, nor uniformly).
Remark 1.5.6. TheL∞ norm kfkL∞(µ) of a measurable function
f: X→Cis defined to the infimum of all the quantitiesM ∈[0,+∞] that are essential upper bounds for f in the sense that |f(x)| ≤ M
for almost everyx. Thenfn converges tof inL∞ norm if and only
ifkfn−fkL∞(µ)→0 asn→ ∞. The L∞ andL1 norms are part of the larger family ofLp norms, studied in§1.3 ofAn epsilon of room, Vol. I.
One particular advantage of L1 convergence is that, in the case
when thefn are absolutely integrable, it implies convergence of the
integrals, Z X fn dµ→ Z X f dµ,
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 conver- gence 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” be- haviour 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 hypothesisµ(X)<∞or auniform integrability hypothesis, then one can obtain some additional implications between the different modes.
1.5.1. Uniqueness. Throughout these notes, (X,B, µ) denotes a measure space. We abbreviate “µ-almost everywhere” as “almost everywhere” throughout.
Even though the modes of convergence all differ from each other, they are all compatible in the sense that they never disagree about
which functionf a sequence of functionsfn converges to, outside of
Proposition 1.5.7. Let fn: X → C be a sequence of measurable functions, and let f, g:X → C be two additional measurable func- tions. Suppose that fn converges to f along one of the seven modes of convergence defined above, andfn converges to g along another of the seven modes of convergence (or perhaps the same mode of con- vergence as for f). Then f andg agree almost everywhere.
Note that the conclusion is the best one can hope for in the case of the last five modes of convergence, since as remarked earlier, these modes of convergence are unaffected if one modifiesf orgon a set of measure zero.
Proof. In view of Exercise 1.5.2, we may assume thatfn converges
to f either pointwise almost everywhere, or in measure, and simi- larly thatfnconverges togeither pointwise almost everywhere, or in
measure.
Suppose first thatfn converges to bothf andgpointwise almost
everywhere. Then by Exercise 1.5.1, 0 converges to f−g pointwise almost everywhere, which clearly implies that f −g is zero almost everywhere, and the claim follows. A similar argument applies if fn
converges to bothf andg in measure.
By symmetry, the only remaining case that needs to be consid- ered is whenfn converges to f pointwise almost everywhere, andfn
converges togin measure. We need to show thatf =galmost every- where. It suffices to show that for everyε >0, that|f(x)−g(x)| ≤ε
for almost everyx, as the claim then follows by settingε= 1/mfor
m = 1,2,3, . . . and using the fact that the countable union of null sets is again a null set.
Fixε > 0, and letA :={x∈ X : |f(x)−g(x)| > ε}. This is a measurable set; our task is to show that it has measure zero. Suppose for contradiction thatµ(A)>0. We consider the sets
AN :={x∈A:|fn(x)−f(x)| ≤ε/2 for alln≥N}.
These are measurable sets that are increasing inN. Asfn converges
to f almost everywhere, we see that almost every x∈ A belongs to at least one of theAN, thusS
∞
set. In particular, µ( ∞ [ N=1 AN)>0.
Applying monotone convergence for sets, we conclude that
µ(AN)>0
for some finite N. But by the triangle inequality, we have |fn(x)−
g(x)| > ε/2 for all x ∈ AN and all n ≥ N. As a consequence, fn
cannot converge in measure tog, which gives the desired contradic-
tion.
1.5.2. The case of a step function. One way to appreciate the distinctions between the above modes of convergence is to focus on the case when f = 0, and when each of the fn is a step function,
by which we mean a constant multiplefn =An1En of a measurable set En. For simplicity we will assume that the An >0 are positive
reals, and that theEn have a positive measure µ(En)>0. We also
assume theAn exhibit one of two modes of behaviour: either theAn
converge to zero, or else they are bounded away from zero (i.e. there exists c > 0 such that An ≥ c for every n. It is easy to see that if
a sequenceAn does not converge to zero, then it has a subsequence
that is bounded away from zero, so it does not cause too much loss of generality to restrict to one of these two cases.
Given such a sequence fn = An1En of step functions, we now ask, for each of the seven modes of convergence, what it means for this sequence to converge to zero along that mode. It turns out that the answer to question is controlled more or less completely by the following three quantities:
(i) Theheight An of thenthfunctionfn;
(ii) Thewidth µ(En) of thenthfunctionfn; and
(iii) The Nth tail support E∗
N :=
S
n≥NEn of the sequence
f1, f2, f3, . . ..
Indeed, we have:
Exercise 1.5.3 (Convergence for step functions). Let the notation and assumptions be as above. Establish the following claims:
(i) fn converges uniformly to zero if and only if An → 0 as
n→ ∞.
(ii) fn converges inL∞ norm to zero if and only ifAn →0 as
n→ ∞.
(iii) fn converges almost uniformly to zero if and only ifAn →0
asn→ ∞, or µ(E∗N)→0 asN → ∞.
(iv) fn converges pointwise to zero if and only if An → 0 as
n→ ∞, orT∞
N=1EN∗ =∅.
(v) fnconverges pointwise almost everywhere to zero if and only
ifAn→0 asn→ ∞, orT∞N=1EN∗ is a null set.
(vi) fn converges in measure to zero if and only if An → 0 as
n→ ∞, orµ(En)→0 asn→ ∞.
(vii) fn converges in L1 norm if and only if Anµ(En) → 0 as
n→ ∞.
To put it more informally: when the height goes to zero, then one has convergence to zero in all modes except possibly for L1 conver-
gence, which requires that the product of the height and the width goes to zero. If instead the height is bounded away from zero and the width is positive, then we never have uniform orL∞convergence, but we have convergence in measure if the width goes to zero, we have almost uniform convergence if the tail support (which has larger measure than the width) has measure that goes to zero, we have pointwise almost everywhere convergence if the tail support shrinks to a null set, and pointwise convergence if the tail support shrinks to the empty set.
It is instructive to compare this exercise with Exercise 1.5.2, or with the four examples given in the introduction. In particular:
(i) In the escape to horizontal infinity scenario, the height and width do not shrink to zero, but the tail set shrinks to the empty set (while remaining of infinite measure throughout). (ii) In the escape to width infinity scenario, the height goes to zero, but the width (and tail support) go to infinity, causing theL1norm to stay bounded away from zero.
(iii) In the escape to vertical infinity, the height goes to infinity, but the width (and tail support) go to zero (or the empty set), causing theL1 norm to stay bounded away from zero.
(iv) In the typewriter example, the width goes to zero, but the height and the tail support stay fixed (and thus bounded away from zero).
Remark 1.5.8.The monotone convergence theorem (Theorem 1.4.44) can also be specialised to this case. Observe that the fn =An1En are monotone increasing if and only if An ≤An+1 and En ⊂En+1
for eachn. In such cases, observe that thefn converge pointwise to
f := A1E, where A := limn→∞An and E :=S∞n=1En. The mono-
tone convergence theorem then asserts that Anµ(En) → Aµ(E) as
n→ ∞, which is a consequence of the monotone convergence theo- remµ(En)→µ(E) for sets.
1.5.3. Finite measure spaces. The situation simplifies somewhat if the spaceX has finite measure (and in particular, in the case when