Lebesgue measure

Overview §

The Lebesgue outer measure for a set $E$, denoted $m^{\ast }$, is defined as the greatest lower bound of the sum of lengths in any open cover of $E$. The Lebesgue measure $m$ is simply the restriction of outer measure to measurable sets.

Outer measure and measure share most key properties, with the important distinction being that measure is countably additive, while outer measure is only countably subadditive. Countable additivity is precisely what gives the Lebesgue integral an advantage over the Properties of the Darboux integral.

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Outer measure §

Definition: Lebesgue outer measure

Given a set $E\subset \mathbb{R}$, let $\{I_k{\}}_{k=1}^{\infty }$ be the collection of all open covers of $E$; that is, each interval $I_k$ is open and bounded. The Lebesgue outer measure of $E$ is the greatest lower bound of the sum of such collections’ lengths.

$$m^{\ast }(E):=\text{inf}\left\{\sum _{k=1}^{\infty }{I}_k|E\subset \bigcup _{k=1}^{\infty }{I}_k\right\}$$

Outer measure has the following key properties:

• Monotonicity. If $F\subset E$ is a subset, then $m^{\ast }(F)\le m^{(}E)$.
• Translation invariance. For all $E\subset \mathbb{R}$ and $x\in \mathbb{R}$, we have $m^{\ast }(E+x)=m^{\ast }(\{z\in \mathbb{R}|z=y+x\text{ for }y\in E\})=m^{\ast }(E).$
• Countable subadditivity. We have $m^{\ast }\left({\bigcup }_{k=1}^{\infty }{E}_k\right)\le {\sum }_{k=1}^{\infty }{m}^{\ast }(E_k).$

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Lebesgue measure and countable additivity §

Definition: Lebesgue measure

The Lebesgue measure is the restriction of the outer measure to the class of measurable sets. If $E$ is a measurable set, its Lebesgue measure is denoted by $m$ and defined as

$$m(E)=m^{\ast }(E):=\text{inf}\left\{\sum _{k=1}^{\infty }{I}_k|E\subset \bigcup _{k=1}^{\infty }{I}_k\right\}.$$

The key difference between properties of outer measure and properties of the Lebesgue measure is that the latter is countably additive.

Proposition: Countable additivity of measure

Given a countable collection of measurable and disjoint sets $\{E_k{\}}_{k=1}^{\infty }$, then their union ${\bigcup }_{k=1}^{\infty }{E}_k$ is also measurable and

$$m\left(\bigcup _{k=1}^{\infty }{E}_k\right)=\sum _{k=1}^{\infty }m(E_k).$$

We can also name the following key properties of measure:

• Finite additivity. Countable additivity holds for any finite disjoint collection of measurable sets $\{E_k{\}}_{k=1}^n$.
• Excision. If $A\subset B$ and $m(A)<\infty$, then $m(B\setminus A)=m(B)-m(A)$.
• Countable monotonicity. This is an oft-used combination of montonicity and countable additivity of measure: if $E$ is covered by a countable collection of measurable sets $\{E_k{\}}_{k=1}^{\infty }$, then

$$m(E)\le \sum _{k=1}^{\infty }m(E_k).$$