Riemann Integral in Rn

Subjects: Vector Analysis
Links: Darboux Sums in Rn, Riemann Integral in R, Jordan Measure

Def: Let $f:R\subseteq {\mathbb{R}}^n\to \mathbb{R}$ bounded over the rectangle $R$. We say that $f$ is Riemann integrable* (or simply integrable)* over $R$ if the lower integral and the upper integral of $f$ over $R$ are equal. In other words

in this case, this number is called ***************the integral of $f$ over $R$ and denoted by

Riemann Criterion of Integrability

Let $f:R\to \mathbb{R}$ bounded over $R$. $f$ is integrable over $R$ with integral $I$ iff for any $\epsilon >0$ there’s a $\delta >0$ such that any partition $P$ into rectangles $R_1,R_2,\dots ,R_N$ with sides $<\delta$ and if $x_1\in R_1,x_2\in R_2,\dots ,x_N\in R_N$ we have that
$$ \left|\sum_{k = 1}^N f(x_i)\cdot m(R_i) - I\right| < \varepsilon $$

Darboux Criterion of Integrability

Let $f:R\to \mathbb{R}$ bounded over $R$. $f$ is Integrable over $R$ iff for all $\epsilon >0$ there’s a partition $\mathcal{P}$ such that

since it can be used a lot the difference of the upper and lower sum. I will use the shorter notation

Th: Let $f:R\to \mathbb{R}$ be continuous over the rectangle $R$. Then $f$ is integrable over $R$.

Lemma: If $f:R\to \mathbb{R}$ is integrable over $R$, and $R^{\prime }$ is a subrectangle of $R$. Then $f$ is integrable over $R^{\prime }$

Lemma: If $f:R\to \mathbb{R}$ be integrable over $R$. Then for any $\epsilon >0$, there’s a partition such that there’s an $i\in \{1,\dots ,k\}$ where $k$ is the number of subrectangles of the partition such that

Th: Let $f:R\to \mathbb{R}$ be integrable over $R$. Then there’s an $x_0\in \mathrm{int}(R)$ such that $f$ is continuous at $x_0$

Def:* Let $f:A\subseteq {\mathbb{R}}^n\to \mathbb{R}$. Then the set $D_{f,A}$ or $D(f,A)$ is the set of all discontinuities of $f$ over $A$, or

Cor: Let $f:R\to \mathbb{R}$ is integrable over $R$. Then $\mathrm{int}(D(f,A)=\varnothing$.

Cor: Let $f:R\to \mathbb{R}$ is integrable over $R$. Then $R\setminus (D(f,A))$ is dense in $R$.

• Algebraic Properties of the Integral
  Th: Let $f,g:R\subseteq {\mathbb{R}}^n\to \mathbb{R}$, then

  • $f+g$ is integrable over $R$ and

    $${\int }_{R}f+g={\int }_{R}f+{\int }_{R}g$$

  • If $c\in \mathbb{R}$, then $cf$ is integrable over $R$ and$$ \int_R cf = c\int_R f $$

  • $f^2$ is integrable over $R$

  • $fg$ is integrable over $R$

  • $|f|$ is integrable over $R$ and $$ \left|\int_R f\right|\le \int_R |f| $$

  • $f\ge 0$ then $$ \int_R f \ge 0 $$

  • $f\ge g$ then $$ \int_R f \ge \int_R g $$

  • the functions $max\{f,g\}$ and $min\{f,g\}$ are integrable over $R$

Prop:****** Let $f:R\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be integrable over $R$ such $f\ge 0$. If $f$ is continuous at $x_0$ and $f(x_0)>0$ then

Prop: Let $f:R\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be integrable over $R$ such that $f>0$. Then

Th: Let $f:R\subseteq {\mathbb{R}}^n\to [a,b]\subset \mathbb{R}$ is integrable over $R$ and $h:[a,b]\to \mathbb{R}$ be continuous, then $h\circ f$ is integrable over $R$.

Mean Value Theorem for Integrals

Let $f,g:R\subseteq {\mathbb{R}}^n\to \mathbb{R}$ such that $f$ is continuous over $R$ and $g$ is integrable over $R$. Then

• ${\int }_{R}f=f(x_0)\cdot m(R)$ for some $x_0\in R$
• if $g\ge 0$ then ${\int }_{R}fg=f(x_0){\int }_{R}g$ for some $x_0\in R$

We can consider Sets of Measure Zero in Rn for Riemann integrability if we need to check when a certain function is Riemann Integrable. We can generalise this integral to functions with Jordan measurable domains.