published 17 January 2018
The usual pedagogical metaphor (from Lebesgue himself) to contrast the Riemann integral with the Lebesgue integral is counting coins. Suppose one has several stacks of coins in front of them, each stack possibly of a different height and possibly containing multiple denominations. The Riemann-integral approach to finding the total value in front of them is to sum each stack, one at a time, and then sum those sums. This follows the definition of the Riemann integral $\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x_i$, in which one finds the area of the first rectangle, then adds the area of the next rectangle, and so on.
The Lebesgue-integral approach, however, is to take apart the stacks; collect all the pennies together, all the nickels together, all the dimes together, etc.; count the number of coins in each heap; multiply by the face-value of the coin in each heap (e.g., if there are 27 nickels, then multiply 27×5 to get the value of the heap); and then sum those values. This is the natural way one would find the total value of coins in a piggy bank.
And it corresponds to the definition of the Lebesgue integral
$\int_E f\,d\mu=\sup\int_E s\,d\mu$
where a simple function
$s(x)=\sum_{i=1}^nc_i1_{E_i}(x)$
is a finite linear combination of indicator functions on measurable sets $E_i$,
its integral $\int_E s\,d\mu:=\sum_{i=1}^nc_i\mu(E_i\cap E)$,
and the supremum is taken over all simple functions $s$ where $0\le s\le f$.
The simple functions play the role of heaping together coins of the same denomination,
since a bunch of plateaus in the graph of $f$ of the same height $c_i$ can be approximated by
a single summand $c_i1_{E_i}(x)$ in a simple function, where $E_i$ is the disjoint union of the bases of the plateaus.
Real analysis instructors and textbooks often say that for Riemann integrals, one partitions the $x$-axis and thus vertically partitions the graph of $f$ into thin oblong rectangles. For Lebesgue integrals, one partitions the $y$-axis and thus horizontally partitions the graph of $f$ into sets of blocks of the same height.
Enough exposition.
While reading James C. Scott’s The Art of Not Being Governed,
I came across another potential metaphor for the Lebesgue integral,
in which area studies and anthropology have benefited from the same horizontal partitioning
perspective
that was fruitful to real analysis.
The book is a study of the stateless, incredibly diverse communities populating the hills of Southeast Asia,
comprising a region called Zomia
, and the dialectic between hill peoples
and the states in the valleys.
Scott argues that the history of the hill peoples is not of those left behind by the development of civilization in the valleys,
but rather of peoples who deliberately escaped the valley states
to avoid taxation, conscription, slavery, and corvée labor,
which were often demanded by the states’ monopoly on violence.
Of course, the valley states – both premodern monarchies and colonial empires – who conquered or wanted to conquer some highland terroritory sought to understand the hill peoples inhabiting them in order to better exploit them.
Early colonial officials, taking an inventory of their new possessions in the hills, were confused to encounter hamlets with several
peoplesliving side by side: hill people who spoke three or four languages and both individuals and groups whose ethnic identity had shifted, sometimes within a single generation. Aspiring to Linnaean specificity in the classification of peoples as well as flora, territorial administrators were constantly frustrated by the bewildering flux of peoples who refused to stay put.^{1}
That is to say, early surveyors tried to analysis the highland population hamlet-by-hamlet or ethnic group-by-ethnic group, thus partitioning the hills by patches of area. This is analogous to the Rieman integral of a function $z=f(x,y)$, in which one partitions the $xy$-plane into squares and thus partitions the space under the graph of $f$ into vertical columns.
The horizontal partitioning
perspective of the Lebesgue integral proved more useful to later anthropologists:
There was, however, one principle of location that brought some order to this apparent anarchy of identity, and that was its relation to altitude. As Edmund Leach originally suggested, once one looks at Zomia not from a high-altitude balloon but, rather, horizontally, in terms of lateral slices through the topography, a certain order emerges. In any given landscape, particular groups often settled within a narrow range of altitudes to exploit the agro-economic possibilities of that particular niche. Thus, for example, the Hmong have tended to settle at very high altitudes (between one thousand and eighteen hundred meters) and to plant maize, opium, and millet that will thrive at that elevation. If from a high-altitude balloon or on a map they appear to be a random scattering of small blotches, this is because they ahve occupied the mountaintops and left the midslopes and intervening valleys to other groups. […] The
Akhaalong the Yunnan-Thai border and theHaniin the upper reaches of the Red River in northern Vietnam are recognizably the same culture, though separated by more than a thousand kilometers. They typically have more in common with each other than either group has with valley people a mere thirty or forty miles away.^{2}
So tell that to your real analysis students when they’re bored.
I wonder where else this 90° rotation in perspective from vertical partitioning
to horizontal partitioning
proves useful, besides integration, coin-counting, and area studies.
★
Scott, James C. The Art of Not Being Governed. New Haven: Yale University Press, 2009, pp. 18–19.^{^}
Ibid.^{^}