Let be a measure space. The measure is said to be absolutely continuous with respect to the measure if we have that for such that (see ). In this case, we denote that is absolutely continuous with respect to by writing .
Recall that if is a measurable function, then the set function for is a measure on . Observe that if , then so that (see  for further details on this example and others).
It was previously established on a homework problem that for some nonnegative measurable defined on the measure space and some arbitrarily chosen , there exists such that whenever (see ). The method that was used to establish this result can also be used to show that, in a finite measure space, if , then for some arbitrarily chosen , there exists such that .
- In particular, we proceed by contradiction and suppose there exists so that for any and , we have . Now, define a sequence of sets such that and denote where . We have from countable subadditivity that . We have from monotonicity that . The monotonicity of the measure implies that . Applying continuity from above to , we also have . However, this contradicts the definition of .
In fact, the converse to the above result also holds (see ). Namely, if we have that there exists so that , then . Suppose that for some . Then, for any such as in the preceding claim, we have . Since can be taken to be arbitrarily small, we have that , as required for the measure to be absolutely continuous with respect to .
: Taylor, M. "Measure Theory and Integration". 50-51.
: Craig, K. "Math 201a: Homework 8". Fall 2020. Refer to question 2.
: Rana, I. K. "Introduction to Measure and Integration". Second Edition. 311-313.