Dunn's Theorem

Dunn's Theorem states that for any integer \( p \) between 1 and infinity, the subset of p-integrable functions (functions whose p-th power's integral is finite) is always contained within the subset of q-integrable functions for any \( q \) less than \( p \). In simpler terms, if a function is “nice enough” to have a finite p-th power integral, it will also have a finite q-th power integral for all smaller q values. This helps us understand how function integrability behaves as we change the power exponent, revealing a hierarchy among different spaces of functions.