In Chapter 3, we already introduced the notion of absolute continuity (see Definition 3.3.19) and then stated an important result, called the Radon-Nikodym Theorem. In this section, we will prove this theorem along with several other important results. Recall that if \(\mu \) and \(\nu \) are measures on a \(\sigma \) -field \(\mathcal{A}\) , then \(\nu \) is said to be absolutely continuous with respect to \(\mu \) , written \(\nu \ll \mu \) , if any \(\mu \) -null set in \(\mathcal{A} \) is also \(\nu \) -null. We start with a simple result that gives an equivalent and useful characterization of absolute continuity of a finite measure \(\nu \) with respect to a measure \(\mu \) . This is often called the ‘ \(\epsilon \) - \(\delta \) criterion’ of absolute continuity.