Exercise 1.1.26 (Higher dimensional Riemann-Darboux integral)

Question

Extend the definition of Riemann-Darboux integral to higher dimensions.

Elu Thingol · Accepted Answer

\begin{definition} (Piecewise constant integral) Let $E \subset \mathbb{R}^d$ be an elementary set, and let $f:E	o\mathbb{R}$ be a piecewise constant function with respect to partition $B_1,\ldots,B_n$ and values $c_1,\ldots,c_n$. Then the piecewise constant integral of $f$ is defined by
$$	ext{p.c.}\int_E f = \sum_{i=1}^{n} c_i \cdot m(B_i)$$
\end{definition}

\begin{definition} (Darboux integral) Let $E \subset \mathbb{R}^d$ be an elementary set, and let $f:E	o\mathbb{R}$ be a bounded function. We define the lower Darboux integral by
$$\underline{\int_E} f := \sup\left\{ 	ext{p.c.}\int_{[a,b]} h: \begin{array}{l} 	ext{$h$ is p.c.}\ h \leq f \end{array} ight\}$$
and the upper Darboux integral by
$$\overline{\int_E} f := \inf\left\{ 	ext{p.c.}\int_{[a,b]} g: \begin{array}{l} 	ext{$g$ is p.c.}\ f \leq g \end{array} ight\}$$
We say that $f$ is Darboux integrable iff both two quantities are equal.
\end{definition}

Exercise 1.1.26 (Higher dimensional Riemann-Darboux integral)

Answers

Comments

Exercise 1.1.26 (Higher dimensional Riemann-Darboux integral)

Answers

Comments

Add answer