# The Unapologetic Mathematician

## Measures

From this point in, I will define a “set function” as a function $\mu$ whose domain is some collection of subsets $\mathcal{E}\subseteq P(X)$. It’s important to note here that $\mu$ is not defined on points of the set $X$, but on subsets of $X$. For some reason, a lot of people find that confusing at first.

We’re primarily concerned with set functions which take their values in the “extended real numbers” $\overline{\mathbb{R}}$. That is, the value of $\mu(E)$ is either a real number, or $+\infty$, or $-\infty$, with the latter two being greater than all real numbers and less than all real numbers, respectively.

We say that such a set function $\mu:\mathcal{E}\to\overline{\mathbb{R}}$ is “additive” if whenever we have disjoint sets $E_1$ and $E_2$ in $\mathcal{E}$ with disjoint union $E_1\uplus E_2$ also in $\mathcal{E}$, then we have

$\displaystyle\mu(E_1\uplus E_2)=\mu(E_1)+\mu(E_2)$

Similarly, we say that $\mu$ is finitely additive if for every finite, pairwise disjoint collection $\{E_1,\dots,E_2\}\subseteq\mathcal{E}$ whose union is also in $\mathcal{E}$ we have

$\displaystyle\mu\left(\biguplus\limits_{i=1}^nE_i\right)=\sum\limits_{i=1}^n\mu(E_i)$

And we say that $\mu$ is countably additive of for every pairwise-disjoint sequence $\{E_i\}_{i=1}^\infty$ of sets in $\mathcal{E}$ whose union is also in $\mathcal{E}$, we have

$\displaystyle\mu\left(\biguplus\limits_{i=1}^\infty E_i\right)=\sum\limits_{i=1}^\infty\mu(E_i)$

Now we can define a “measure” as an extended real-valued, non-negative, countably additive set function $\mu$ defined on an algebra $\mathcal{A}$, and satisfying $\mu(\emptyset)=0$. With this last assumption, we can show that a measure is also finitely additive. Indeed, given a collection $\{E_1,\dots,E_n\}$, just define $E_i=\emptyset$ for $i>n$ to get a sequence. Then we find

$\displaystyle\mu\left(\biguplus\limits_{i=1}^nE_i\right)=\mu\left(\biguplus\limits_{i=1}^\infty E_i\right)=\sum\limits_{i=1}^\infty\mu(E_i)=\sum\limits_{i=1}^n\mu(E_i)$

If $\mu$ is a measure on $\mathcal{A}$, we say a set $A\in\mathcal{A}$ has finite measure if $\mu(A)<\infty$. We say that $A$ has “$\sigma$-finite” measure if there is a sequence of sets $\{A_i\}_{i=1}^\infty$ of finite measure ($\mu(A_i)<\infty$) so that $A\subseteq\bigcup_{i=1}^\infty A_i$. If every set in $A$ has finite (or $\sigma$-finite) measure, we say that $\mu$ is finite (or $\sigma$-finite) on $A$.

Finally, we say that a measure is “complete” if for every set $A$ of measure zero, $\mathcal{A}$ also contains all subsets of $A$. That is, if $A\in\mathcal{A}$, $\mu(A)=0$, and $B\subseteq A$, then $B\in\mathcal{A}$. At first, this might seem to be more a condition on the algebra $\mathcal{A}$ than on the measure $\mu$, but it really isn’t. It says that to be complete, a measure can only assign ${0}$ to a set if all of its subsets are also in $\mathcal{A}$.

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Posted by John Armstrong | Analysis, Measure Theory

## 17 Comments	»

1. This stuff is SO important, and so rarely done well. I’m confident that you can make it delightful.

Comment by | March 20, 2010 | Reply

2. thanks for these excellent posts, i been trying to teach my self measure theory i’m finding these posts very useful.
looking forward to the rest

Comment by learningToMeasure | March 20, 2010 | Reply

3. [...] I say that any measure on an algebra is both monotone and subtractive. Indeed, since is an algebra, then is guaranteed [...]

Pingback by | March 22, 2010 | Reply

4. [...] I say that a measure is continuous from above and [...]

Pingback by | March 23, 2010 | Reply

5. [...] as Metric It turns out that a measure turns its domain into a sort of metric space, measuring the “distance” between two [...]

Pingback by | March 24, 2010 | Reply

6. [...] We’re going to want a modification of the notion of a measure. But before we introduce it, we have (of course) a few [...]

Pingback by | March 25, 2010 | Reply

7. [...] a Measure to an Outer Measure Let be a measure in a ring (not necessarily an algebra) , and let be the hereditary -ring generated by . For every [...]

Pingback by | March 26, 2010 | Reply

8. [...] measure on a hereditary -ring is nice and all, but it’s not what we really want, which is a measure. In particular, it’s subadditive rather than additive. We want to fix this by restricting to [...]

Pingback by | March 29, 2010 | Reply

9. [...] additive. That is, if we define for , then is actually a measure. Even better, it’s a complete [...]

Pingback by | March 30, 2010 | Reply

10. [...] and Induced Measures If we start with a measure on a ring , we can extend it to an outer measure on the hereditary -ring . And then we can [...]

Pingback by | March 31, 2010 | Reply

11. [...] of a Measure At last we can show that the set function we defined on semiclosed intervals is a measure. It’s clearly real-valued and non-negative. We already showed that it’s monotonic, and [...]

Pingback by | April 16, 2010 | Reply

12. [...] We’ve spent a fair amount of time discussing rings and -rings of sets, and measures as functions on such collections. Now we start considering how these sorts of constructions relate [...]

Pingback by | April 26, 2010 | Reply

13. [...] we don’t particularly care if the set where is false is itself measurable, although if is complete then all -negligible sets will be measurable. This sort of language is so common in measure theory [...]

Pingback by | May 13, 2010 | Reply

14. [...] If is a.e. non-negative, then will also be non-negative, and so the indefinite integral is a measure. Since is integrable we see [...]

Pingback by | June 21, 2010 | Reply

15. [...] We continue what we started yesterday by extending the notion of a measure. We want something that captures the indefinite integrals of every function for which it’s [...]

Pingback by | June 22, 2010 | Reply

16. [...] I say that each of these set functions — , , and — is a measure, and that . If is (totally) finite or -finite, then so are and , and at least one of them will [...]

Pingback by | June 25, 2010 | Reply

17. [...] real-valued functions on them. Given a function on a Boolean ring , we say that is additive, or a measure, -finite (on -rings), and so on analogously to the same concepts for set functions. We also say [...]

Pingback by | August 5, 2010 | Reply

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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