# Thread: Riemann integral is zero for certain sets

1. ## Riemann integral is zero for certain sets

The question is:

Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, then $\int_{E}f(x)dV=0$.

(How to relate the condition it's Riemann integrable to the value is $0$? The textbook I use define $f$ is integrable on $E$ iff $\;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV$)

2. ## Re: Riemann integral is zero for certain sets

Originally Posted by ianchenmu
The question is:

Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, then $\int_{E}f(x)dV=0$.

(How to relate the condition it's Riemann integrable to the value is $0$? The textbook I use define $f$ is integrable on $E$ iff $\;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV$)
What is the (L) and (U)? In any case, since the integral exists, it must be zero since the set you're integrating over has Lebesgue measure zero.

3. ## Re: Riemann integral is zero for certain sets

Originally Posted by Gusbob
What is the (L) and (U)? In any case, since the integral exists, it must be zero since the set you're integrating over has Lebesgue measure zero.
$(L)\int_{E}fdV=\underset{G}{sup}L(f;G)$,

$(U)\int_{E}fdV=\underset{G}{inf}U(f;G)$,

$U(f;G)=\sum_{R_j\cap E\neq \O }M_j|R_j|$ where $M_j=sup_{x\in R_j}f(x)$,
$L(f;G)=\sum_{R_j\cap E\neq \O }m_j|R_j|$ where $m_j=inf_{x\in R_j}f(x)$.

( $E$ is an Jordan region in $\mathbb{R}^n$, $f$ is a bounded function, $R$ is an n-dimensional rectangle s.t. $E\subseteq R$.
$G=\left \{ R_1,...,R_p \right \}$ is a grid on $R$. (Extend $f$ to $\mathbb{R}^n$ by setting $f(x)=0$ for $x\in\mathbb{R}^n\setminus E$.))

Can you use these backgrounds and terminologies to explain? Thanks!