Riemann Integral in R

Subjects: Real Analysis
Links: Riemann and Darboux Sums in R

Integrability in Riemann

Original Definition

Let $f:I\to \mathbb{R},$ be Reimann integrable on $I$ if and only if there:

$L$ is also called the Reimann integral of $f$ along $I$, also denoted as: ${\int }_{I}f$ . The set of Reimann integrable functions is denoted as: ${\mathcal{R}}_I$

Other notation is: $\underset{\parallel \dot{P}\parallel \to 0}{lim}R(f,\dot{P})=L$ .

Theorem: If $f\in {\mathcal{R}}_I$, then the integral is uniquely determined.

Theorem: If $f\in {\mathcal{R}}_I$, and $f=g$ except at finitely many points, then $g\in {\mathcal{R}}_{\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}}$ and .

Theorem: If $f\in {\mathcal{R}}_I$, then $f$ is bounded on $I$

Cauchy Criterion

Let $f:I\to \mathbb{R},$ be Reimann integrable on $I$ if and only if there:

Squeeze Theorem

Let $f:I\to \mathbb{R},$ be Reimann integrable on $I$ if and only if:

Integrability

Definition

Let $f:I\to \mathbb{R}$, is Darboux integrable if and only if $L(f)=U(f)$, and the Darboux Integral of $f$ over $I$:

Integrability Criterion

Let $f:I\to \mathbb{R}$, is Darboux integrable if and only if:

Sequential Criterion

Let $f:I\to \mathbb{R}$, is Darboux integrable if and only if there exists a sequence of partitions $({\mathcal{P}}_n)\subseteq {\mathrm{\wp }}_I$ such that:

Equivalence

If a function is Reimann Integrable if and only if it is Darboux Integrable

General type of Integrable functions: ${\mathcal{C}}^0(I)\subset {\mathcal{R}}_I$, and all monotonic functions since they are continuous almost everywhere. ${\mathcal{R}}_I$ is an associative commutative algebra over $\mathbb{R}$.

Additivity Theorem

******Let $f$ be integrable in $[a,b]$ if and only if $f$ is integrable in $[a,c]$ and $[c,b]$ for some $c\in (a,b)$, and:

Corollary: $[c,d]\subseteq [a,b]⟹{\mathcal{R}}_{[c,d]}\subseteq {\mathcal{R}}_{[a,b]}$

Definition: If $a<b$ then:

Algebraic Properties

Suppose that $f,g\in {\mathcal{R}}_I$, and $c\in \mathbb{R}$, then:

1. $cf\in {\mathcal{R}}_I$, and ${\int }_{I}cf=c{\int }_{I}f.$
2. $f+g\in {\mathcal{R}}_I$, and ${\int }_I(f+g)={\int }_{I}f+{\int }_{I}g$.
3. If $m\le f\le M⟹m\mu (I)\le {\int }_{I}f\le M\mu (I)$, where $\mu$ is the Lebesgue measure.
4. If $f\le g$ on $I⟹{\int }_{I}f\le {\int }_{I}g$.
5. The function $|f|\in {\mathcal{R}}_I$ , and $|{\int }_{I}f|\le {\int }_I|f|$.

Lebesgue’s Criterion

Let $f:[a,b]\to \mathbb{R}$, $f\in {\mathcal{R}}_{[a,b]}⟺\mu (D_f)=0$, where $\mu$ is the Lebesgue Measure, and $D_f$ is the set of all discontinuities of $f$, or $f$ is continuous almost everywhere on $[a,b]$.

Composition Theorem

Let $f\in {\mathcal{R}}_{[a,b]}$, and $f([a,b])\subseteq [c,d]$ , where $\phi \in {\mathcal{C}}^{\mathcal{0}}[c,d]$, then $\phi \circ f\in {\mathcal{R}}_{[a,b]}$

Corollary:

If $f,g\in {\mathcal{R}}_I⟹fg\in {\mathcal{R}}_I$

Interchange of limit and integral

Integrable limit theorem

Assume that $f_n\rightrightarrows f$ on $I$, and that each $f_n\in {\mathcal{R}}_I$, then $f\in {\mathcal{R}}_I$ and that:

Bounded Convergence Theorem

Assume that $f_n\mathbb{R}ightarrowf$, and $\mathrm{\exists }K>0\mathrm{\forall }n\in \mathbb{N}(|f_n|\le K)$, with both , $\mathrm{\forall }n\in \mathbb{N}(f_n\in {\mathcal{R}}_I),f\in {\mathcal{R}}_I$, then:

Theorem

Let $g_n\to g$ on $[a,b]$ and uniformly on $[a,\gamma ]$ where $\gamma \in (a,b)$, and $g_n$ is uniformly bounded and integrable, then:

Let $f:[a,b]\to \mathbb{R}$ be bounded on $[a,b]$ and for any $c\in (a,b]$, $f\in {\mathcal{R}}_{[c,b]}$. Then, $f\in {\mathcal{R}}_{[a,b]}$ and,