Publicacions Ma em`a iques, Vol 39 (1995), 273–283.
WEIGHTED LpSPACES
AND POINTWISE ERGODIC THEOREMS
Ryo a o Sa o
Abs ac
In his pape we gi e an ope a o heo e ic e sion o a ecen e-
sul o F. J. Ma ´ın-Reyes and A. de la To e conce ning he p ob-
lem o finding necessa y and sufficien condi ions o a nonsingula
poin ans o ma ion o sa is y he Poin wise E godic Theo em in
Lp. We conside a posi i e conse a i e con ac ion Ton L1o
aσ-fini e measu e space (X, F,µ), a fixed unc ion ein L1wi h
e>0onX, and wo posi i e measu able unc ions Vand Won
X. We hen cha ac e ize he pai s (V,W) such ha o any in
Lp(Vdµ) he a e ages
Rn
0( ,e)= n

k=0
Tk  n

k=0
Tke
con e ge almos e e ywhe e o a unc ion in Lp(Wdµ). The cha -
ac e iza ions a e gi en o all p,1≤p<∞.
1. In oduc ion
Le (X,F,µ)beaσ-fini e measu e space and Ta posi i e linea con-
ac ion o L1(µ). We assume T o be a conse a i e ope a o . (Fo he
usual no a ion we e e he eade o K engel’s book [2].) Thus he class
(1) I=I(T)={A∈F:T∗1A=1
A}
o all in a ian se s ela i e o T o ms a σ-field, whe e 1Adeno es he
indica o unc ion o Aand T∗deno es he adjoin ope a o o T, ac ing
on L∞(µ). Since Tis posi i e, we may ex end by a canonical manne he
domain o T o he class M+(µ) o all nonnega i e ex ended eal alued
measu able unc ions on X. Simila ly, his is done o T∗. Now le us fix
an e∈L1(µ) wi h e>0onX. Le 0 <V,W≤∞be wo measu able
274 R. Sa o
unc ions on X. P e iously we obse ed in [6] ha i 1 <p<∞ hen
o any ∈L+
p(Vdµ) he a e ages
(2) Rn
0( ,e)=n

k=0
Tk  n

k=0
Tke
con e ge o a fini e limi a.e. on Xi and only i
(3) E{e−1V1−p|(X,I,edµ)}<∞a.e. on X,
whe e 1/p +1/p=1. In[6] we also obse ed implici ely (see especially
p. 76–77 in [6]) ha o any ∈L+
1(Vdµ) he a e ages Rn
0( ,e) con e ge
o a fini e limi a.e. on Xi and only i he e exis s a unc ion U,
measu able wi h espec o I, such ha
(4) V−1≤U<∞a.e. on X.
In his pape we in end o s udy he p oblem o cha ac e ing he
case whe e he limi unc ion R∞
0( ,e) belongs o L+
p(Wdµ) o e e y
∈L+
p(Vdµ). This s udy was inspi ed by he wo k [3] o Ma ´ın-Reyes
and de la To e. See also Assani and W´os [1]. As a esul , his pape
may be conside ed o be an ope a o heo e ic e sion o Ma ´ın-Reyes
and de la To e’s pape [3]. Using he esul ob ained we nex conside
mul ipa ame e poin wise e godic heo ems o commu ing posi i e lin-
ea con ac ions o L1(µ) ha ing a common s ic ly posi i e fixed poin
in L1(µ).
2. The main esul
Theo em 1. Le Tbe a conse a i e posi i e linea con ac ion o
L1(µ).Le V,Wbe wo posi i e eal alued measu able unc ions on X.
Fix an e∈L1(µ)wi h e>0on X.I 1<p<∞and 1/p +1/p=1,
hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(Vdµ) he a e ages Rn
0( ,e)con e ge a.e. o a
unc ion belonging o L+
p(Wdµ).
(b) E{e−1W|(X,I,edµ)}1/p ·E{e−1V1−p|(X,I,edµ)}1/p≤Ca.e.
on X, whe e Cis a posi i e cons an .
(c) Fo any ∈L+
p(W1−pdµ) he a e ages Rn
0( ,e)con e ge a.e.
o a unc ion belonging o L+
p(V1−pdµ).
I p=1, hen (a) is equi alen o
(d) E{e−1W|(X,I,edµ)}≤CV a.e. on X.
Poin wise e godic heo ems 275
P oo : Le 1 <p<∞.
(a) ⇒(b). By (a) he limi unc ion
(5) R∞
0( ,e) = lim
nRn
0( ,e)
is fini e a.e. on X. Thus by (3) we ha e
E{e−1V1−p|(X,I,edµ)}<∞a.e. on X.
Choose Xn∈I,n=1,2,..., so ha
(6) Xn↑Xand Xn
V1−pdµ < ∞.
Since V(1−p)p·V=V1−p, i ollows ha
(7) V1−p∈L+
p(Xn,V dµ).
On he o he hand, since R∞
0(·,e) is a posi i e linea ope a o om
Lp(Vdµ)in oLp(Wdµ) by (a), i is bounded, i.e., he e exis s a cons an
K>0 such ha
(8) |R∞
0( ,e)|pWdµ≤Kp| |pVdµ ( ∈Lp(Vdµ)).
The e o e, o any A∈Iwi h A⊂Xn, (7) yields
(9) A
R∞
0(V1−p,e)pWdµ≤KpA
V1−pdµ < ∞.
Since R∞
0(V1−p,e)=E{e−1V1−p|(X,I,edµ)}a.e. on X(c . p. 73 in
[6]), hese inequali ies imply
R∞
0(V1−p,e)pE{e−1W|(X,I,edµ)}≤KpE{e−1V1−p|(X,I,edµ)}
<∞a.e. on X;
and (b) ollows.
(b) ⇒(a). Since e−1 =(e−1/p V1/p)(e−1/pV−1/p), he H¨olde in-
equali y o he condi ional expec a ion ope a o and (b) imply ha i
∈L+
p(Vdµ) hen
R∞
0( ,e)=E{e−1 |(X,I,edµ)}
≤E{e−1 pV|(X,I,edµ)}1/p ·E{e−1V1−p|(X,I,edµ)}1/p
≤CE{e−1 pV|(X,I,edµ)}1/p ·E{e−1W|(X, I,edµ)}−1/p
276 R. Sa o
a.e. on X; and hus
R∞
0( ,e)pWdµ≤CpE{e−1 pV|(X,I,edµ)}
E{e−1W|(X,I,edµ)}Wdµ
=CpE{e−1 pV|(X,I,edµ)}edµ
=Cp pVdµ<∞,
which p o es (a).
(b) ⇔(c). Di ec om (a) ⇔(b).
Le p=1.
(a) ⇔(d). Fo any ∈L+
1(Vdµ) we ob ain
R∞
0( ,e)Wdµ=E{e−1 |(X,I,edµ)}Wdµ
=E{e−1 |(X,I,edµ)}E{e−1W|(X,I,edµ)}edµ
= E{e−1W|(X,I,edµ)}dµ
= V(E{e−1W|(X,I,edµ)}·V−1)dµ.
Hence, by (8) wi h p= 1, (a) is equi alen o
(a) V(E{e−1W|(X,I,edµ)}·V−1)dµ ≤K V dµ o e e y
∈L+
1(Vdµ);
and (a) is clea ly equi alen o (d). The p oo is comple e.
Co olla y 1. In addi ion o he hypo heses o Theo em 1, i we as-
sume ha Tis e godic, i.e., ha Iis i ial, hen he ollowing a e
equi alen , o e e y 1≤p<∞:
(a) Fo any ∈L+
p(Vdµ) he a e ages Rn
0( ,e)con e ge a.e. o a
unc ion belonging o L+
p(Wdµ).
(b) W∈L1(µ)and V−1∈Lp(Vdµ), whe e p=∞when p=1.
3. Applica ions
Le d≥1 be an in ege and T1,... ,T
dbe commu ing posi i e linea
con ac ions o L1(µ). In his sec ion we assume ha he e exis s an
e∈L1(µ) wi h e>0onXsuch ha
(10) Tie=e(1 ≤i≤d).
Poin wise e godic heo ems 277
Thus each Tiis a conse a i e ope a o and sa isfies he mean e godic
heo em in L1(µ). And by an induc ion a gumen we see ha o any
∈L1(µ) he a e ages
(11) An(T1,... ,T
d) =An(T1)...A
n(Td)
con e ge in L1-no m, whe e An(Ti)= 1
n
n−1

k=0
Tk
i. By Theo em 1 o [5],
o any ∈L1(µ) he a e ages An(T1,... ,T
d) con e ge a.e. on X. Le
us deno e he limi unc ion by A(T1,... ,T
d) ; hus
(12) A(T1,... ,T
d) = lim
nAn(T1,... ,T
d) a.e. on X.
I we le
(13) T=1
d
d

i=1
Ti
hen Talso sa isfies he mean e godic heo em in L1(µ); and we ge he
di ec decomposi ion
L1(µ)={ ∈L1(µ):T = }⊕{g−Tg :g∈L1(µ)}−.
Since T = i and only i Ti = o each 1 ≤i≤dby he B unel-
Falkowi z lemma (c . p. 82 in [2]) and
lim
nAn(T1,... ,T
d)(g−Tg)1=0
by he equa ion g−Tg =1
d
d

i=1
(g−Tig), i ollows ha o any ∈L1(µ)
he limi unc ion A(T1,... ,T
d) coincides a.e. wi h he limi unc ion
(14) A(T) = lim
nAn(T) .
Fu he , since I(T)=
d

i=1
I(Ti) (in he sequel Iwill deno e his σ-field),
i ollows ha o any ∈L+
1(µ)
A(T1,... ,T
d) = lim
nAn(T)
= lim
nen

k=0
Tk  n

k=0
Tke
(15)
=eE{e−1 |(X, I,edµ)}a.e. on X.
Hence, by an app oxima ion a gumen , o any ∈M+(µ) he limi
A(T1,... ,T
d) = lim
nAn(T1,... ,T
d) exis s a.e. on Xand sa isfies (15).
We a e now in posi ion o s a e he fi s applica ion o Theo em 1.

278 R. Sa o
Theo em 2. Le T1,... ,T
dbe commu ing posi i e linea con ac ions
o L1(µ)such ha Tie=e(1 ≤i≤d) o some e∈L1(µ)wi h e>0on
X.Le 0<V,W<∞be wo measu able unc ions on X.I 1<p<∞
and 1/p +1/p=1, hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(Vdµ) he limi unc ion A(T1,... ,T
d) belongs
o L+
p(Wdµ).
(b) E{ep−1W|(X,I,edµ)}1/pE{e−1V1−p|(X,I,edµ)}1/p≤Ca.e.
on X.
(c) Fo any ∈L+
p(e−pW1−pdµ) he limi unc ion A(T1,... ,T
d)
belongs o L+
p(e−pV1−pdµ).
I p=1, hen (a) is equi alen o
(d) E{W|(X,I,edµ)}≤CV a.e. on X.
Consequen ly, in case Iis i ial, (a) is equi alen , o e e y
1≤p<∞, o
(e) epW∈L1(µ)and V−1∈Lp(Vdµ), whe e p=∞when p=1.
P oo : Fo any ∈L+
p(Vdµ) we ha e, by (15), A(T1,... ,T
d) =
eR∞
0( ,e). Thus (a) is equi alen o
(a)Fo any ∈L+
p(Vdµ) he limi unc ion R∞
0( ,e)( ela i e o T)
belongs o L+
p(epWdµ).
The e o e, by Theo em 1, we see (a) ⇔(b) when 1 <p<∞, and
(a) ⇔(d) when p= 1. When 1 <p<∞, (b) ⇔(c) ollows om he
equi alence (a) ⇔(b). This comple es he p oo .
Co olla y 2. Le T1,... ,T
dand ebe he same as in Theo em 2. I
1<p<∞and 1/p +1/p=1, hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(µ) he limi unc ion A(T1,... ,T
d) belongs o
L+
p(µ).
(b) E{ep−1|(X,I,edµ)}1/pE{e−1|(X,I,edµ)}1/p≤Ca.e. on X.
Consequen ly, in case Iis i ial, (a) is equi alen , o e e y
1<p<∞, o
(c) µ(X)<∞and e∈Lp(µ).
Rema k. We no e ha (a) o Co olla y 2 always holds when p=1.
We nex conside he adjoin ope a o s T∗
1,... ,T∗
d. Since
(T∗
i )edµ = (Tie)dµ = edµ o ∈L+
∞(µ), T∗
1,... ,T∗
dcan be
ega ded as commu ing posi i e linea con ac ions o L1(edµ). Since
(16) T∗
i1=1∈L1(edµ)(1≤i≤d),
Poin wise e godic heo ems 279
i we eplace he measu e µand he unc ion eby edµ and 1, espec-
i ely, hen he abo e-gi en a gumen shows ha o any ∈M+(µ)=
M+(edµ) he limi
(17) A(T∗
1,... ,T∗
d) = lim
nAn(T∗
1,... ,T∗
d)
exis s a.e on X; u he , since I=
d

i=1
I(Ti)=
d

i=1
I(T∗
i), i ollows ha
(18) A(T∗
1,... ,T∗
d) = lim
nAn(T∗) =E{ |(X,I,edµ)}a.e. on X,
whe e T∗=1
d
d

i=1
T∗
i.
Theo em 3. Le T1,... ,T
dand ebe he same as in Theo em 2. Le
0<V,W<∞be wo measu able unc ions on X.I 1<p<∞and
1/p +1/p=1, hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(Vdµ) he limi unc ion A(T∗
1,... ,T∗
d) belongs
o L+
p(Wdµ).
(b) E{e−1W|(X,I,edµ)}1/pE{(e−1V)1−p|(X,I,edµ)}1/p≤Ca.e.
on X.
(c) Fo any ∈L+
p(epW1−pdµ) he limi unc ion A(T∗
1,... ,T∗
d)
belongs o L+
p(epV1−pdµ).
(d) Fo any ∈L+
p(W1−pdµ) he limi unc ion A(T1,... ,T
d) be-
longs o L+
p(V1−pdµ).
(e) Fo any ∈L+
p(e−pVdµ) he limi unc ion A(T1,... ,T
d) be-
longs o L+
p(e−pWdµ).
I p=1, hen (a) is equi alen o
( ) E{e−1W|(X,I,edµ)}≤C(e−1V)a.e. on X.
Consequen ly, in case Iis i ial, (a) is equi alen , o e e y
1≤p<∞, o
(g) W∈L1(µ)and eV −1∈Lp(Vdµ), whe e p=∞when p=1.
P oo : Since L+
p(Vdµ)=L+
p(e−1Vedµ) and L+
p(Wdµ)=
L+
p(e−1Wedµ), i we apply Theo em 2 o commu ing posi i e linea
con ac ions T∗
1,... ,T∗
do L1(edµ), hen (16) yields (a) ⇔( ) when
p= 1, and (a) ⇔(b) ⇔(c) when 1 <p<∞. I we w i e (b) as
E{ep−1(e−pW)|(X,I,edµ)}1/pE{e−1(e−pV)1−p|(X,I,edµ)}1/p
≤Ca.e. on X,
280 R. Sa o
and apply Theo em 2 o commu ing posi i e linea con ac ions
T1,... ,T
do L1(µ), hen we ob ain (b) ⇔(e) ⇔(d) when 1 <p<∞.
The p oo is comple e.
Co olla y 3 (c . [3] and [4]). Le T1,... ,T
dand ebe he same as
in Theo em 2. I 1<p<∞and 1/p +1/p=1, hen he ollowing a e
equi alen :
(a) Fo any ∈L+
p(µ) he limi unc ion A(T∗
1,... ,T∗
d) belongs o
L+
p(µ).
(b) E{e−1|(X,I,edµ)}1/pE{ep−1|(X,I,edµ)}1/p≤Ca.e. on X.
(c) Fo any ∈L+
p(µ) he limi unc ion A(T1,... ,T
d) belongs o
L+
p(µ).
I p=1, hen (a) is equi alen o
(d) E{e−1|(X,I,edµ)}≤Ce−1a.e. on X.
Consequen ly, in case Iis i ial, (a) is equi alen , o e e y
1≤p<∞, o
(e) µ(X)<∞and e∈Lp(µ), whe e p=∞when p=1.
Co olla y 4. Suppose (X,F,µ)is a fini e measu e space. Le
T1,... ,T
dbe commu ing posi i e linea con ac ions o L1(µ), and as-
sume ha µis in a ian unde T1,... ,T
d, i.e., ha Ti1=1∈L1(µ)
(1 ≤i≤d).Le 0<V,W<∞be wo measu able unc ions on X.I
1<p<∞and 1/p +1/p=1, hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(Vdµ) he limi unc ion A(T∗
1,... ,T∗
d) belongs
o L+
p(Wdµ).
(b) E{W|(X,I,µ)}1/pE{V1−p|(X,I,µ)}1/p≤Ca.e. on X.
(c) Fo any ∈L+
p(W1−pdµ) he limi unc ion A(T∗
1,... ,T∗
d)
belongs o L+
p(V1−pdµ).
I p=1, hen (a) is equi alen o
(d) E{W|(X,I,µ)}≤CV a.e. on X.
Consequen ly, in case Iis i ial, (a) is equi alen , o e e y
1≤p<∞, o
(e) W∈L1(µ)and V−1∈Lp(Vdµ), whe e p=∞when p=1.
Rema k. Unde he hypo heses o Co olla y 4, i ollows (see (15)
and (18)) ha o any ∈M+(µ)
A(T1,... ,T
d) =A(T∗
1,... ,T∗
d) =E{ |(X,I,µ)}a.e. on X,
so ha he unc ion A(T∗
1,... ,T∗
d) can be eplaced by he unc ion
A(T1,... ,T
d) in Co olla y 4, wi hou any influence.
Poin wise e godic heo ems 281
4. Concluding ema ks
Th oughou his sec ion, (X, F,µ)isaσ-fini e measu e space, and
T1,... ,T
da e commu ing posi i e linea con ac ions o L1(µ) such ha
Tie=e(1 ≤i≤d) o some e∈L1(µ) wi h e>0onX. He e we b iefly
discuss he p oblem o cha ac e izing a posi i e measu able unc ion V
on Xsuch ha i ∈L+
p(Vdµ) hen he limi unc ion A(T1,... ,T
d)
(o A(T∗
1,... ,T∗
d) ) is fini e a.e. on X. As in he p eceding sec ion, we
will deno e I=
d

i=1
I(Ti). The esul s may be s a ed as ollows. (Fo a
ela ed esul we e e he eade o [7].)
Theo em 4. Le 0<V ≤∞be a measu able unc ion on X.I
1<p<∞and 1/p +1/p=1, hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(Vdµ) he limi unc ion A(T1,... ,T
d) is fini e
a.e. on X.
(b) E{e−1V1−p|(X,I,edµ)}<∞a.e. on X.
I p=1, hen (a) is equi alen o
(c) V−1≤U<∞a.e. on X o some U, measu able wi h espec o
I.
P oo : By i ue o (15) and he esul men ioned in In oduc ion (see
especially (3) and (4)), Theo em 4 ollows immedia ely.
Co olla y 5. I 1<p<∞, hen he ollowing a e equi alen :
(a) Fo any ∈L+
p(µ) he limi unc ion A(T1,... ,T
d) is fin ie a.e.
on X.
(b) The e exis Xn∈I,n=1,2,..., such ha Xn↑Xand µ(Xn)<
∞.
(c) Fo any ∈
1≤ ≤∞
L+
(µ) he limi unc ion A(T1,... ,T
d) is
fini e a.e. on X.
P oo : Since he implica ions (b) ⇒(c) ⇒(a) a e ob ious, we only
p o e (a) ⇔(b). To do his we apply Theo em 4 wi h V=1onXand
see ha (a) is equi alen o
E{e−1|(X,I,edµ)}<∞a.e. on X,
which is clea ly equi alen o (b). The p oo is comple e.