= Average value of a function
{title2=$\overline f_I=|I|^{-1}\int_I f(x)\,dx$}
= Cell average
{synonym}
For an <integrable function> on an <interval> $I$ of positive finite length, its average value is the <integral> divided by that length:
$$
\overline f_I=\frac1{|I|}\int_I f(x)\,dx.
$$
If $f$ is constant on equal-length subintervals, this is their values' <arithmetic mean>; unequal lengths require a <weighted mean>. For example, values $2$ and $6$ on two equal halves give average value $4$, whereas value $2$ on the first quarter and $6$ on the remaining three quarters give $5$. The <integral> measures total contribution; the <average value of a function> divides out the interval length.
For the <function> $f(x)=2x$ on $[0,1]$, the <average value of a function> is $\int_0^1 2x\,dx=1$. Replacing it by the constant $1$ preserves its <integral> on this interval. This is why the <Haar projection> uses <cell averages> to obtain a coarser description while retaining each cell's total contribution.
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