As a part of my measure theory class each student must lecture on a certain topic. Today I lectured on the Jordan decomposition and the Radon-Nikodym Theorem. Lecturing to freshman precalculus students is one thing, lecturing to your peers and professor about measure theory is another. You encourage precalc students to ask questions and be engaged with the lecture. As the teacher you try to make the lecture more of a discussion. This is something you can do because you know and understand the material. However, this can be difficult when you don’t fully understand the subject (like measure theory). The other TA’s and I like to joke about our measure theory lectures, fake it ’til you make it. We might not have a mastery understanding of the material (yet), but we have the fundamentals. From there it’s all about looking and feeling confident when delivering your lecture.* *

For those who are interested, these are my notes on the lecture I gave today.* The Elements of Integration and Lebesgue Measure* by Robert Bartle is the reference for specific theorems, corollaries, and lemmas.

**Definition.** Let be a charge on and let be a Hahn decomposition for The *positive* and the *negative variations* of are the finite measures , defined for in by , The *total variation* of is the measure defined for in by

Note, by lemma 8.3, and are well defined and do not depend on the Hahn decomposition.

**Theorem 8.5 Jordan Decomposition Theorem.** If is a charge on then for all in Moreover, if where and are finite measures on then and for all in

*Proof.* We prove the first assertion. Let and let be a Hahn decomposition of Then and We have,

We prove the second assertion. Since and have nonnegative values,

Similarly,

**Theorem 8.6.** If is integrable, , and is defined by then

*Proof. *Let and . Then and . Let . Then

Thus, is a Hahn decomposition of with respect to . Now, by the definition of variation we have that

and the theorem is established.

**Definition. **A measure on is said to be *absolutely continuous *with respect to a measure on if and imply that . In this case we write . A charge is* absolutely continuous* with respect to a charge provided that the total variation of is absolutely continuous with respect to .

**Example. **Let and . Recall, is a measure by corollary 4.9. If for some then –almost everywhere. Thus,

which shows that is absolutely continuous with respect to .

**Example.** Let be Lebesgue measure and the counting measure on . Then if and only if . Hence, implies that , which shows that . However, if , the singleton set, then but . Thus, is not absolutley continuous with respect to .

Absolute continuity can also be characterized as follows.

**Lemma 8.8.** Let and be finite measures on . Then if and only if for every there exists a such that and imply that .

*Proof.* If for any then this is necessary and sufficient for . Conversely, suppose that there exists and sets with and . Let so that and . Since is a decreasing sequence of measurable sets, we have that

Hence, does not imply that .

Recall, corollary 4.9 states that if then is a measure. Conversely, when can we express a measure as an integral with respect to of some function ?

Corollary 4.11 showed that a necessary condition for this to hold is that . It turns out that this condition is also sufficient when and are -finite. We state this result as a theorem.

**Theorem 8.9 Radon-Nikodym Theorem. **Let amd be -finite measures on with . Then there exists such that for . Moreover, the function is uniquley determined -almost everywhere.

## Leave a comment

Comments feed for this article