= Density of continuous functions in Lp on an interval
{title2=$\overline{C([a,b])}^{\|\cdot\|_p}=L^p((a,b)),\quad1\leq p<\infty$}
For a finite interval $(a,b)$ with <Lebesgue measure>, restrictions of <continuous functions> on $[a,b]$ form a <dense subspace> of the <Lp space> $L^p((a,b))$ for $1\leq p<\infty$. One proof first approximates by <simple functions>. The <regularity of Lebesgue measure> approximates each of their <measurable sets> in measure by a finite union of intervals. An interval's <indicator function> is approximated by <continuous functions> taking values in $[0,1]$ and differing from that <indicator function> only in endpoint strips of total length tending to zero. Their <Lp norm> error is at most the $1/p$ power of that length. The <triangle inequality> completes the approximation of each <simple function>.
Back to article page
