A Generalization of Holder's Inequality and Some Probability Inequalities

The main result of this article is a generalization of the generalized Holder inequality for functions or random variables defined on lower-dimensional subspaces of $n$-dimensional product spaces. It will be seen that various other inequalities are included in this approach. For example, it allows the calculation of upper bounds for the product measure of $n$-dimensional sets with the help of product measures of lower-dimensional marginal sets. Furthermore, it yields an interesting inequality for various cumulative distribution functions depending on a parameter $n \in \mathbb{N}$.