Lee Tuo-Yeong
Abstract
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It is shown that if $f$ is Henstock-Kurzweil integrable on a compact interval $E$ in ${\mathbb R}^m$, then $f$ is upper semicontinuously integrable on $E$ if and only if there exists an increasing sequence $\{X_n\}$ of closed sets whose union is $E$, and $f |_{X_n}$ is bounded for each positive integer $n$.
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It is shown that if $f$ is Henstock-Kurzweil integrable on a compact interval $E$ in ${\mathbb R}^m$, then $f$ is upper semicontinuously integrable on $E$ if and only if there exists an increasing sequence $\{X_n\}$ of closed sets whose union is $E$, and $f |_{X_n}$ is bounded for each positive integer $n$.
Key concepts: Mathematics, Integrable system, Locally integrable function, Bounded function, Interval (graph theory), Characterization (materials science), Integer (computer science), Sequence (biology)