172   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

AMERICAN Journal of Engineering,  
Mechanics and Architecture 

Volume 2, Issue 3, 2024    ISSN (E): 2993-2637 

 
  

Isometries of log -Integrable Functions 
 

Abdullaev R. Z. 

Doctor of Physical and Mathematical Sciences, professor. Tashkent University of information 

Technologies 

Madaminov B. A. 

Doktor of Physical and Mathematical ciences (Phd). Non-government educational institution 

"Mamun University" 

 

Abstract: In this paper studied isometries of F spaces of  integrable functions with logarithm. 

In the paper, it is given isomorphic classification of F -spaces of log -integrable measurable 

functions constructed using different measures. At the same time, it is proved that such spaces 

are non-isometric.  

Keywords: Functional spaces; boolean algebras; complete boolean algebras; homogeneous 

Boolean algebras; internal log algebras; external log algebras; generalized log algebras; 

isometries. 

 

 

1. Introduction 

One of the important classes of Banach functional spaces are spaces ( , , ),pL  A  1 <p   

of all p -th power integrable measurable functions given on the measurable space ( , , ) A  

with the finite measure   (almost everywhere equal functions are identified). The study of 

isometries of Banach spaces pL  was initiated by S. Banach [1], who gave a description of all 

isometries for spaces [0,1],pL  2p  . In [2], J. Lamperti gave characterization of all linear 

isometries for pL -spaces ( , , )pL  A  where ( , , ) A  is an arbitrary space with the finite 

measure .  The final result in this setting is due to Yeadon [3] who gave a complete description 

of all isometries between pL -spaces associated with different measures. One of the corollaries 

of such descriptions of isometries in spaces ( , , )pL  A  is the establishment of isometry for 

pL -spaces ( , , )pL  A  and ( , , )pL  A  in the case when the measures   and   are strictly 

positive finite measures. 

An important metrizable analogue of Banach spaces pL  are F -spaces log ( , , )L  A  of log -

integrable measurable functions introduced in the work [4]. An F -space log ( , , )L  A  is 

defined by the equality  

 log 0( , , ) = { ( , , ) : log(1 | |) < }L f L f d  


    A A  



 

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where 0 ( , , )L  A  is the algebra of all measurable functions given on ( , , ) A  (almost 

everywhere equal functions are identified). By virtue of the inequality  

 
1

log( ( )) ( ) , , [1, ),pf f p
p

       

the inclusion ( , , ) ( , , )p logL L   A A  is always true. 

In [4], it was established that ( , , )logL  A  is a subalgebra in the algebra 0 ( , , ).L  A  In 

addition, a special F -metric ( , )f g  has been introduced in ( , , ),logL  A   

 log( , ) = log(1 | |) , , ( , , ).f g f g d f g L  


    A  

The pair log( ( , , ), )L   A  is a complete metric topological vector space with respect to this 

measure, and the operation of multiplication f g  is continuous in the totality of variables. 

It is evident, algebras 0 ( , , )L  A  and 
0( , , )L  A  coincide when the measures   and   are 

strictly positive finite measures. This fact is no longer true for the algebras ( , , )logL  A  and 

( , , ).logL  A  In the work [5], it was shown that ( , , ) = ( , , )log logL L  A A  for strictly 

positive finite measures   and   if and only if ( , , )
d

L
d





  A  and ( , , ),

d
L

d





  A  

where ( , , )L   A  is the algebra of all essentially bounded measurable functions given on 

( , , ) A  (almost everywhere equal functions are identified), 
d

d




 (respectively, 

d

d




) is the 

Radon-Nikodym derivation of the measure   (respectively,  ) with respect to the measure   

(respectively,  ). 

Isometries on these F -spaces were considered in [6]. In these papers a description of isometries 

on F -spaces was given. In contrast to these results, in this paper we establish a necessary and 

sufficient condition for the existence of isometries and isomorphisms of algebras of log-

integrable functions constructed by different measures   and .  The relationship between these 

isometries and isomorphisms is also studied. In this case, conditions are imposed only on the 

Radon-Nikodym derivatives .
d

d




 

Naturally the problem arises to find the necessary and sufficient conditions providing an 

isomorphism of the algebras ( , , )logL  A  and ( , , )logL  A  for the strictly positive finite 

measures   and .  The solution of this problem is given in section 4. 

The main purpose of this paper is to prove the absence of surjective isometries from 

( , , )logL  A  onto ( , , )logL  A  in the case when measures   and   are strictly positive 

finite measures (see section 4). 

 

2. Preliminaries 

Let ( , , ) A  be a be a complete measure space with finite measure   and let 0 ( , , )L  A  

(respectively, ( , , )L   A ) be the algebra of equivalence classes of real valued measurable 



 

174   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

functions (respectively, essentially bounded real valued measurable functions) on ( , , ) A . 

Denote by =    the complete Boolean algebra of all equivalence classes =[ ],e A  ,AA  

of equal  -almost everywhere sets from the  -algebra A . It is known that 

( ) = ([ ]) = ( )e A A    is a strictly positive finite measure on .  In what follows, we also 

denote the measure   by ,  and the algebra 0 ( , , )L  A  (respectively, ( , , )L   A ) by 

0 0( ) = ( )L L    (respectively, ( ) = ( )).L L     

Following to [4], consider in 0 ( )L   a subalgebra  

 log 0( ) = { ( ) : log(1 | |) < }L f L f d  


      

of log -integrable measurable functions, and for each log ( ),f L    set  

 log = log(1 | |) .f f d


P P  

By [4, Lemma 2.1], a nonnegative function log log: ( ) [0, )L    P P  is a F -norm, that is, 

( )i . log > 0fP P  for all log0 ( );f L     

( )ii . log logf f P P P P  for all log ( )f L    and real number   with | | 1  ; 

( )iii . 0 log = 0lim f  P P  for all log ( );f L    

( )iv . log log logf g f g  P P P P P P  for all log, ( ).f g L    

In [4] it is shown that log ( )L   is a complete topological algebra with respect to the topology 

generated by the metric log( , ) = .f g f g P P  

Let   and   be two strictly positive measures on the measurable space ( , ) A . Then  

 0 0 0( ) = ( ) = ( ), ( ) = ( ) = ( ).L L L L L L            

Let 
d

d




 be the Radon-Nikodym derivative of measure   with respect to the measure  . It is 

well known that 00 ( )
d

L
d




    and  

 1 1( , , ) ( , , ),
d

f L f L
d


 


     A A  

in addition,  

 = ( ) .
d

f d f d
d


 


 

   

Note that for strictly positive measures   and  , it follows that 
1( ) = .

d d

d d

 

 


 



 

175   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

If measure   is finite, then 1( )
d

L
d






  . This follows from the equality ( ) = ( ) <h  1 . 

3  Isometries of the F -spaces  log ( )L    and  log ( )L   

 

In this section, a necessary and sufficient condition for the existence of isometries is established 

of ( , , )logL  A  onto ( , , ).logL  A  

Let   and   strictly positive finite measures,  =
d

h
d




, and let  

 

1

0( ) = { ( ) : = ( | | ) < };p p

p pL f L f f d  


   P P  

 

 

1 1

0( ) = { ( ) : = ( | | ) = ( | | ) < }.p pp p

p pL f L f f d h f d   
 

     P P  

In this case the map : ( ) ( )p pU L L    , defined by the following equality  

 

1

( ) = , ( ),p

pU f h f f L 



   

is the non trivial surjective isometry from ( )pL   onto ( )pL  . 

Bellow we show that this statement is not true for F -spaces log ( )L   and log ( )L  . 

Note that  

 log 0( ) = { ( ) : log(1 | |) < } =L f L f d 


      

 

 0= { ( ) : log((1 | |) < } =f L h f d


      

 

 0= { ( ) : log((1 | |) ) < }hf L f d


     

and  

 , = (1 | |) = log((1 | |) < } =logf log f d h f d  
 

    P P  

 

 = log((1 | |) ) .hf d


  



 

176   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

Let   be a non-atomic complete Boolean algebra that is, a Boolean algebra   has not atoms. 

Let = { : },e g g e    where 0 e  . By ( )e   denote the minimal cardinality of a 

set that is dense in 
e  with respect to the order topology ( ( )o -topology). The non-atomic 

complete Boolean algebra   is said to be homogeneous if ( ) = ( )e g    for any nonzero 

, .e g  The cardinality ( )   is called the weight of the homogeneous Boolean algebra   

(see, for example [7, chapter VII]). 

 

Theorem 3.1  Let   be a complete homogeneous Boolean algebra,   and   be finite strictly 

positive measures on .  ( )logL   is isometric to ( )logL   iff =1.
( )

hd







 

  

Proof. Necessity is proved in [5], see Theorem 5. We prove sufficiency. Let = 1
( )

hd







. Then 

using the equality 
1= = ( ),hd h hd  

 

   where 
1 =

d
h

d






, we obtain ( ) = ( )   . 

Hence, there is a measure-preserving automorphism   from   onto 
 , i.e. 

( ) = ( ( ))e e    for any e   ([7, chapter VII, Theorems 5 and 6]). Denote by J  the 

isomorphism of the algebra 0 0( ) = ( )L L    such that ( ) = ( )J e e   for all e  . That’s 

why for any ( ),logf L    we have from ([5], Proposition 3)  

 , = (1 | ( ) |) = ( (1 | ( ) |)) =logf log f d J log f d    
 

  P P  

 

 ,= (1 | ( ) |) = | ( ) | .loglog J f d J f    


 P P  

Hence, the J  is a bijective linear isometry from ( )logL   onto ( )logL  .  

 

Let =
d

h
d




 and 

> < = > <={ : ( ) >1}, ={ : ( ) <1}, ={ : ( ) =1}= \ ( ).h h h             

 Denote: 

 > >= ( , ) =S S h  >

>

( ( ) 1)

( )

h x d










 and < <= ( , ) =S S h  <

<

(1 ( ))

.
( )

h x d










 The following 

theorem establishes 5 conditions equivalent to the isometricity of F-spaces. 

 



 

177   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

Theorem 3.2 Let   be a complete homogeneous algebra,   and   be finite strictly positive 

measures on ,  then the following conditions are equivalent 

( ).i     ( )logL   and ( )logL   are isometric; 

( )

( ). =1;
( )

h d

ii

 







  

1( )

( ). =1,
( )

h d

iii

 










 where 

1 = ;
d

h
d






  

( ). ( ) = ( );iv      

> <( ). =v S S ; 

( ).vi     there is a measure-preserving automorphism   from   onto .   

  

Proof. ( ) ( )i ii  follows from Theorem 1.  

( ) ( )ii iii  

1 1( ) ( ) ( )
( )

= = =1.
( ) ( )

h d h h d

d h d

    


   

 

 

 





 

 
 The reverse implication is 

proved similarly.  

( ) ( )iii iv  

1 1( ) ( ) ( )
( )

= = =1 ( ) = ( )
( ) ( )

h d h h d

d

    


 
 

 

 




  

 

 


  

( ) ( )iv vi  follows [8, chapter VII, Theorems 5 and 6]  

( ) ( )ii v  

>

>

< =
> < > <

< =

( ( ) 1)( ) ( ) ( ( ) 1)

0 = 1 = = =
( ) ( ) ( ) ( ) ( )

( ( ) 1) ( ( ) 1)

= =
( ) ( )

h dh d h d d h d

h d h d

S S S S

       

    

   

 


   

 



  
    

 

  
 

   

 
  

References  

1. S. Banach S, Theorie des operations lineaires. Warsaw, 1932. 

2. J. Lamperti, On the isometries of some function spaces. Pacific J. Math., 8 (1958), 459–466. 

3. F.J. Yeadon, Isometries of non-commutative Lp -spaces. Math. Proc.Camb. Phil. Soc. 90 

(1981) 41-50. 



 

178   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

4. K. Dykema, F. Sukochev, D. Zanin, Algebras of log-integrable functions and operators. 

Complex Anal. Oper. Theory 10 (8) (2016), 1775–1787. 

5. R.Z. Abdullaev, V.I. Chilin, Isomorphic Classification of  -Algebras of Log-Integrable 

Measurable Functions. Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 

2017. Springer Proceedings in Mathematics and Statistics, 264, 73-83. Springer, Cham. 

6. R.Abdullaev, V.Chilin, B.Madaminov Isometric F-spaces of log-integrable function. 

Siberian Electronic Mathematical Reports. том 17,стр. 218-226(2020). 

7. D.A. Vladimirov,  Boolean Algebras in Analysis. Mathematics and its Applications, 540, 

Kluwer Academic Publishers, Dordrecht (2002).