Commutativity of integral quasi-arithmetic means on measure spaces

• Published in: Acta mathematica Hungarica Vol. 153; no. 2; pp. 350 - 355
• Main Authors: Głazowska, D., Leonetti, P., Matkowski, J., Tringali, S.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2017; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; 39B52; 91B99; primary 26E60; functional equation; 39B22; generalized (quasi-arithmetic) mean; commuting mapping; secondary 28E99; 60B99
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-017-0734-2

Let ( X , L , λ ) and ( Y , M , μ ) be finite measure spaces for which there exist A ∈ L and B ∈ M with 0 < λ ( A ) < λ ( X ) and 0 < μ ( B ) < μ ( Y ) , and let I ⊆ R be a non-empty interval. We prove that, if f and g are continuous bijections I → R + , then the equation f - 1 ( ∫ X f ( g - 1 ( ∫ Y g ∘ h d μ ) ) d λ ) = g - 1 ( ∫ Y g ( f - 1 ( ∫ X f ∘ h d λ ) ) d μ ) is satisfied by every L ⊗ M -measurable simple function h : X × Y → I if and only if f  =  cg for some c ∈ R + (it is easy to see that the equation is well posed). An analogous, but essentially different result, with f and g replaced by continuous injections I → R and λ ( X ) = μ ( Y ) = 1 , was recently obtained in [ 7 ].