integrable

Def 28.5 pg. 228 Darboux integrable

For $I=[a,b]$ and $f:I⟶\mathbb{R}$ be a bounded function. Then $f$ is Darboux integrable if $(\exists \tau )(\forall \epsilon >0){\overline{\sum }}_{\tau }f-{\underline{\sum }}_{\tau }f<\epsilon$

Def 28.1 Riemann integrable

$$(\exists I)(\forall \epsilon >0)(\exists \delta )(\exists \sigma )(\exists \xi )|\sigma |<\delta ⟹|R(f,\xi )-I|<\epsilon$$

where $\sigma$ is a partition, $\delta$ is the max mesh size, and $\xi$ is a set with points from every sub-interval in $\sigma$, $R$ is Riemann Sum.

Thm 28.4

Darboux integrable imply Riemann integrable

Thm 29.5

$f$ continue imply $f$ integrable

Lemma 10.4

let $\Phi$ be Lipschitz and $f$ integrable, we have $\Phi \circ f$ integrable.
Also ${\Delta }_{\sigma }(\Phi \circ f)\le L{\Delta }_{\sigma }(f)$

Thm 31.1 Lebesgue’s Criterion for Riemann Integrability

$f$ is bounded and $\exists k\subseteq [a,b]$, the following are equivalent:

1. $f$ is Riemann integrable on $k$
2. the set of discontinuities of $f$ on $k$ has Lebesgue measure zero
   Basically, it means, $f$ is only discontinue in measure zero set, and continue otherwise.

Thm 31.2

$\phi$ is continue, $f$ integrable $⟹$ $\phi \circ f$ integrable

OH 35.1

Thm A

$\phi$ Lipschitz and $f$ integrable $⟹$ $\phi \circ f$ integrable

Thm B

$\phi$ continue and $f$ integrable $⟹$ $\phi \circ f$ integrable

Proof

By Thm C, we can say there exists $S$ has Lebesgue Zero Measure.

Thm C

$f$ integrable $⟺$ $(\exists S)$ has Lebesgue Zero Measure