Absolutely Continuous Measures

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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 [1]). 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 [3] 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 [2]). 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 [3]). 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 .


[1]: Taylor, M. "Measure Theory and Integration". 50-51.

[2]: Craig, K. "Math 201a: Homework 8". Fall 2020. Refer to question 2.

[3]: Rana, I. K. "Introduction to Measure and Integration". Second Edition. 311-313.