Real Analysis I: Measure

This series of articles aim to give a fast review of real analysis and give readers some references for further reading. For the first part, interested readers can open the book "Measure Theory" written by Paul Halmos to find more details. Other recommended reading materials are "Real and Complex Analysis" written by Walter Rudin and "A Course in Functions of Real Variable" by Lihua Yang.

Lets start with the most important concept of the so-called Real Analysis: measure. Roughly speaking, given an arbitrary set , a measure on is a function defined on a subset of , namely the power set of , such that it behaves like "area". The specific subset of is called a -algebra. Sets in the given -algebra are called measurable.

Example 1 Consider the classical example of a measure defined on the Euclidean space $mathbb{R}^n$ . It is called a Borel measure, which takes value at any cube in $mathbb{R}^n$ the volume of this cube and other compactible sets which will be union, intersection, or complement of cubes. Note that cube plays a decisive role in our procedure of defining a measure. Such a set of measurable sets generates a -algebra and hence we can only define the value of a measure on this smaller set to obtain the whole measure.

Now we pick measurable sets such that they have measure and find that in Example 1 above, a subset of a measurable set of measure is not necessarily a measurable set. It is out of our expectation since a subset of sets with area should be has area as well, so we throw them into our -algebra of Borel sets and obtained an enlarged -algebra. We denote this by and call it the set of Lebesgue measurable sets and the measure extending to a Lebesgue measure. This procedure--throwing sets of measure zero into a -algebra--is called completion. Note that any -algebra has a completion, and hence we can always assume a -algebra is complete.

Now we have obtained a measure defined on some complete -algebra defined on some set . We pick them together to construct a measure space or measurable space(here these two terminologies are not the same, but not so much difference) . Now we can discuss integration on this measure space. A function , where is an arbitrary Banach space, is called measurable if the preimage of any open subsets of is measurable. Pick any vector , we can define the integration of $g(x)=leftlbraceegin{array}{cl} v,&xin M\ 0,&x otin M end{array} ight.$ as the vector , where is measurable. The integration of on will be the limit of such simple integrations, where we pick approximating . We use the familiar notation $int_X fmathrm{d}mu$ to denote such an integration. A measurable function is called integrable if the integration $int_XVert fVertmathrm{d}mu <infty$ .

Example 2 We consider again the Euclidean space $mathbb{R}^n$ with the Lebesgue measure defined on it. Assume $fin C_c (mathbb{R}^n )$ is a continuous function with compact support, then the integration $int_{mathbb{R}^n } fmathrm{d} m$ will be the Riemann integration of , so we can calculate it explicitly by indefinite integral or Wolfram Alpha. Since a compact set will be contained in some cube, we have that must be integrable. We denote the space of all integrable functions on by , or simply if it will not make any ambiguity.

The next article of this series will talk about convergence theorems for Lebesgue integral, which tells us why Lebesgue integral attaches analysts attention.