Composite Submeasures and Supermeasures with Applications to Classical Inequalities

Let Ω be a nonempty set. A subset A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3514.tif"/> of the power set 2^Ω is called an algebra if the following conditions are satisfied:

Ω is in A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3515.tif"/> ;

A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3516.tif"/> is closed under complementation, namely, if A ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3517.tif"/> then Ω \ A ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3518.tif"/> ;

A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3519.tif"/> is closed under union, i.e. if A, B ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3520.tif"/> then A ∪ B ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq3521.tif"/> .