Lebesgue measure

[[concept]]

Topics

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Definition

Lebesgue Measure

The Lebesgue measure of a measurable set $E\subset \mathcal{M}$ is given by $m(E)=m^{\ast }(E)$

Where $m^{\ast }(\cdot )$ is the outer measure (see below)

Outer measure

Outer Measure

For $A\subset \mathbb{R}$, we define the outer measure of $A$ $m^{\ast }(A)=\inf \left\{{\sum }_n\ell (I_n):\{I_n\}\text{ countable and open s.t. }A\subset {\bigcup }_n{I}_n\right\}$ We can see that $m^{\ast }(A)\ge 0$ for all $A$.

Example

Consider the set $A=\{0\}$ containing a single point. $m^{\ast }(\{0\})=0$.

Let $ϵ>0$. Then

\{ 0 \} &\subset \left( -\frac{\epsilon}{2} , \frac{\epsilon}{2}\right) \\ \implies m^*(\{ 0 \}) &\leq \ell\left( -\frac{\epsilon}{2}, \frac{\epsilon}{2} \right) = \epsilon \\ \implies m^*(\{ 0 \}) &=0 \end{align}$$ Thus the (outer) measure is 0! ^example

References

References

See Also

Mentions

Mentions

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