Integrable Functions

[Kurt]

I visited Caltech last Thursday through Sunday as part of their Pre-Frosh Weekend (which rocked!), and I sat in on the freshman level math class (Math 1c).

Here are some questions posed there and shortly afterward...see if you can solve them. The discussion was mostly over (duh) integrable functions.

State whether these functions can be integrated (and give the solution if you can find it)...and why:

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[unparseable or potentially dangerous latex formula]


And if [unparseable or potentially dangerous latex formula],

[unparseable or potentially dangerous latex formula]

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In determining negligible areas of functions, are there any curves that have a non-negligible area inherently?


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Evaluate [unparseable or potentially dangerous latex formula]

[makashi]

[unparseable or potentially dangerous latex formula]

This function is not Riemann integrable. Because the rationals are dense in the real numbers, the upper Riemann sum is always [unparseable or potentially dangerous latex formula], while the lower Riemann sum is always [unparseable or potentially dangerous latex formula]. Since the limits are not equal, the function is not Riemann integrable. However, it is Lebesgue integrable. Because the rationals form a countable set of points, it has measure zero. Their complement, the set of irrational numbers therefore has measure one. However, the function is zero at an irrational number. Thus, the Lebesgue integral comes out to zero.


And if [unparseable or potentially dangerous latex formula],

[unparseable or potentially dangerous latex formula]

This function is Riemann integrable. The lower Riemann sum is always zero. Now, we want to show that the greatest lower bound of the upper Riemann sum is zero. This is the same as showing the upper Riemann sum can be less than [unparseable or potentially dangerous latex formula] for any [unparseable or potentially dangerous latex formula]. So, pick [unparseable or potentially dangerous latex formula]. By the Archimedean property of the real numbers, there exists an integer [unparseable or potentially dangerous latex formula] such that [unparseable or potentially dangerous latex formula].

Make the following partition: For any rational number [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula], isolate the point with an interval whose width becomes arbitrarily small. Clearly, there will only be a finite number, say [unparseable or potentially dangerous latex formula], of these intervals. Let the widths of the intervals around the rational numbers with denominator less than [unparseable or potentially dangerous latex formula] be less than [unparseable or potentially dangerous latex formula]. Let's call these intervals A-type intervals. Include each remaining connected part of the unit interval in the partition. Call these intervals B-type intervals.

The upper Riemann sum of the B-type intervals is less than [unparseable or potentially dangerous latex formula].

Now, calculate the upper Riemann sum of the A-type intervals. Because there are [unparseable or potentially dangerous latex formula] intervals, each with width less than [unparseable or potentially dangerous latex formula] and height less than [unparseable or potentially dangerous latex formula], the upper Riemann sum of these A-type intervals is also less than [unparseable or potentially dangerous latex formula].

Thus, the total upper Riemann sum of the partition is less than [unparseable or potentially dangerous latex formula], so the function is Riemann integrable, equal to zero.

Note the same Lebesgue integrable one-line argument still works. ;)