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Let $I\subseteq\mathbb{R}^n$, $P$ a partition of $I$, $f:I\to\mathbb{R}$ a function, and $J$ the indexing set of $P$. Note that each subbox $I^{(\vec{\alpha})}$ is compact, which means that if $f$ is continuous, the minimum and maximum values are attained on each subbox. Then for each $\vec{\alpha}\in J$ we define $m^{(\vec{\alpha})}:=\inf_{\vec{x}\in I^{(\vec{\alpha})}} \{f(\vec{x})\}$ and $M^{(\vec{\alpha})}:=\sup_{\vec{x}\in I^{(\vec{\alpha})}}\{f(\vec{x})\}$ such that
$$\begin{align*}
m^{(\vec{\alpha})}\leq f(\vec{x})\leq M^{(\vec{\alpha})}
\end{align*}$$ for all $\vec{x}\in I^{(\vec{\alpha})}$. Then we define the lower Riemann sum of $f$ with respect to $P$ as
$$\begin{align*}
L(f, P):=\sum_{\vec{\alpha}\in J}m^{(\vec{\alpha})}\mu(I^{(\vec{\alpha})})
\end{align*}$$. We define the upper Riemann sum of $f$ with respect to $P$ as
$$\begin{align*}
U(f, P):=\sum_{\vec{\alpha}\in J}M^{(\vec{\alpha})}\mu(I^{(\vec{\alpha})}).
\end{align*}$$
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Let $I\subseteq\mathbb{R}^n$, $P$ a partition of $I$, $f:I\to\mathbb{R}$ a function, and $J$ the indexing set of $P$. Note that each subbox $I^{(\vec{\alpha})}$ is compact, which means that if $f$ is continuous, the minimum and maximum values are attained on each subbox. Then for each $\vec{\alpha}\in J$ we define $m^{(\vec{\alpha})}:=\inf_{\vec{x}\in I^{(\vec{\alpha})}} \{f(\vec{x})\}$ and $M^{(\vec{\alpha})}:=\sup_{\vec{x}\in I^{(\vec{\alpha})}}\{f(\vec{x})\}$ such that
$$\begin{align*}
m^{(\vec{\alpha})}\leq f(\vec{x})\leq M^{(\vec{\alpha})}
\end{align*}$$ for all $\vec{x}\in I^{(\vec{\alpha})}$. Then we define the lower Riemann sum of $f$ with respect to $P$ as
$$\begin{align*}
L(f, P):=\sum_{\vec{\alpha}\in J}m^{(\vec{\alpha})}\mu(I^{(\vec{\alpha})})
\end{align*}$$. We define the upper Riemann sum of $f$ with respect to $P$ as
$$\begin{align*}
U(f, P):=\sum_{\vec{\alpha}\in J}M^{(\vec{\alpha})}\mu(I^{(\vec{\alpha})}).
\end{align*}$$
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Coming soonUsed In
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FullPage
definition
concepts
used in
hypothesis
results