Lebesgue Differentiation Theorem

Hardy-Littlewood Maximal Function

Vitali's Covering Lemma

Suppose is a finite collection of open balls in some metric space . Then there exists a disjoint sub-collection of such that the -times expansions of satisfy

Corollary

Suppose is a finite collection of open balls in . Then there exists a disjoint sub-collection of that satisfies

Hardy-Littlewood Maximal Function

Suppose is integrable, we define its Hardy-Littlewood maximal function by In other words, is the supremum over all balls containing of the average value of over .

e.g. Hardy-Littlewood maximal function of the characteristic function of the interval is

Theorem

Suppose is integrable. Then the Hardy-Littlewood maximal function is Lebesgue measurable and for almost every .

Theorem

Suppose is integrable. Then there holds for all .

Remark

Broadly speaking, this means is not much larger than .

Lebesgue Differentiation Theorem

Locally Integrable Function

A measurable function is locally integrable if for every ball , the function is integrable.

Proposition

Clearly, if is integrable, then is locally integrable.

Lebesgue Differentiation Theorem

If is locally integrable on , then

Density

Lebesgue Density

Suppose . The density of at is defined as if the limit exists. If the limit does not exist, we say that the density of at is undefined.

e.g. This definition captures the notion of the proportion of a set in small intervals centered at a number . For example, the density of is

Lebesgue Density Theorem

Suppose is a measurable subset of , then the density of is at almost every point of and at almost every point of .

Proof This is a direct consequence of the Lebesgue Differentiation Theorem. To see this, note that if is measurable, then is integrable. Thus, by the Lebesgue Differentiation Theorem, we have This implies that for almost all .

Lebesgue Points

Suppose is locally integrable on . A point is called a Lebesgue point of if is finite and We call the set of Lebesgue points of the Lebesgue set of .

Proposition

If is locally integrable on , then is a Lebesgue point of if and only if is continuous at .

Corollary

If is locally integrable on , then almost every point of is a Lebesgue point of .

Shrink Regularity

A collection of sets is said to shrink regularly to if there is a constant such that for each , there is a ball such that

e.g. In with , the collection of all open rectangles containing does not shrink regularly to . This can be seen if we consider very thin rectangles.

Proposition

Good Kernels and Approximations to the Identity