A new inequality for the Riemann-Stieltjes integrals driven by irregular signals in Banach spaces.

We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces, and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any [Formula: see text], we introduce the space of regulated signals [Formula: see text] ([Formula: see text] are real numbers, and is a Banach space) that may be uniformly approximated with accuracy [Formula: see text] by signals whose total variation is of order [Formula: see text] as [Formula: see text] and prove that they satisfy the assumptions of the theorem. Finally, we derive more exact, rate-independent characterisations of the irregularity of the integrals driven by such signals.