What is an Integral? - Exploring Riemann, Lebesgue & More

[johnqwertyful]

I was explaining to a friend about some differences between the Riemann integral and Lebesgue integral. He asked what an integral is and the best I could come up with was "it defines a notion of area" but I don't think that's good enough. There's Riemann integral, Lebesgue, Darboux, Haar, line, surface, double, multiple, etc. So in its most general case, what is an integral? I haven't been able to find widely accepted axiomatic treatment of integration. Could I just say that the integral of all functions is 0 and call it the "johnqwertyful integral"? Or call any object or function I can think of an integral?

 

[Simon Bridge]

An integral is a fancy way of doing a sum.

It does not "define a notion of area" - it is a way of calculating areas by dividing them up into arbitrarily small bits.

 

[pwsnafu]

I haven't been able to find widely accepted axiomatic treatment of integration.

See the Daniell integral.

 

[johnqwertyful]

An integral is a fancy way of doing a sum.

It does not "define a notion of area" - it is a way of calculating areas by dividing them up into arbitrarily small bits.

My real analysis professor said integrals define area. If not, then how IS area defined? What if two different integrals give you two different areas, which one is the "true" area?

Also, "fancy way of doing sum" is incredibly handwavey. Can I call any fancy way of doing sum an integral? Johnqwertyful integral: add up the natural numbers 1 through 100 then subtract 4. That's an integral because it's a fancy way of doing a sum.

 

[johnqwertyful]

See the Daniell integral.

Interesting. I like the axiomatic treatment of it. How widely are these axioms accepted/used? Are there any interesting examples of integrals that follow these axioms?

 

[WannabeNewton]

An integral is a fancy way of doing a sum.

It does not "define a notion of area" - it is a way of calculating areas by dividing them up into arbitrarily small bits.

Actually a sum is a special type of integral.

 

[mathwonk]

an integral is essentially a way to assign a number m(f) to each function f in some linear space of functions which is linear, i.e. such that m(af+bg) = a.m(f) + b.m(g), and non negative, i.e. such that m(f) ≥ 0 whenever f ≥ 0. I.e. it is a non negative linear functional defined on a linear space of functions.

Different ways of assigning this number m(f) to f are valid for different spaces of functions. E.g. the Riemann and Lebesgue definitions of the number m(f) agree for those functions where both assignments make sense, but the Lebesgue definition applies to a larger space of functions.

If you want to pin it down more you can assume, for functions defined on subsets of the real line, that if f is a function equal to 1 on any interval [a,b] and equal to zero elsewhere, that m(f) = b-a. Just these simple requirements already force the definition of the integral for Riemann integrable functions, i.e. those with sufficiently good approximations from above and below by step functions.

If we add a condition on the behavior of the number m(f) under taking pointwise limits of functions, it seems we can also force the definition of the number m(f) for all Lebesgue integrable functions.

 

[Simon Bridge]

My real analysis professor said integrals define area. If not, then how IS area defined? What if two different integrals give you two different areas, which one is the "true" area?

If two different integrals give you different areas - then how can an integral define "area"?
(Your real-analysis prof was probably being glib.)

Notice that the concept of area in geometry predates the invention of integral calculus and can be taught to students before they know about integration. OTOH: See WannabeNewton's comment below - maybe grade-schoolers have "actually" been doing a form of integration when they thought they were "merely" finding areas?

Also, "fancy way of doing sum" is incredibly handwavey. Can I call any fancy way of doing sum an integral? Johnqwertyful integral: add up the natural numbers 1 through 100 then subtract 4. That's an integral because it's a fancy way of doing a sum.

Naturally. There are lots of different ways to do sums, integration is one of them, but not all ways of doing sums are integration. Integration is closer to grade-school summation than, say, multiplication.

I realize you want some sort of axiomatic approach - you have been given one.

Actually a sum is a special type of integral.

Chickens and eggs.

 

[johnqwertyful]

If two different integrals give you different areas - then how can an integral define "area"?
(Your real-analysis prof was probably being glib.)

You said that integrals calculate area, but what if two different integrals give you two different areas? You're saying that there is one true area that every integral should give you. If that is true, why are their different integrals?

It's the same idea as different metrics. A metric on a set doesn't calculate distance, it defines distance. And two different metrics will give you two different distances.

Notice that the concept of area in geometry predates the invention of integral calculus and can be taught to students before they know about integration. OTOH: See WannabeNewton's comment below - maybe grade-schoolers have "actually" been doing a form of integration when they thought they were "merely" finding areas?

And I'm sure grade-schoolers have been doing addition and multiplication without the slightest idea of what rings, fields, groups, etc. are. Newton did calculus incredibly hand wavey, and when the founders of analysis came through, they formalized certain concepts. Everything gets done hand wavey at first, then gets formalized later.

Naturally. There are lots of different ways to do sums, integration is one of them, but not all ways of doing sums are integration. Integration is closer to grade-school summation than, say, multiplication.

I realize you want some sort of axiomatic approach - you have been given one.

Mhmm, and I appreciate it.

 

[Simon Bridge]

You said that integrals calculate area, but what if two different integrals give you two different areas?

No, I didn't.
I said that integrals are a way of calculating areas.

You're saying that there is one true area that every integral should give you. If that is true, why are their different integrals?

Because there are different size areas and different ways of applying the concept of area.

It's the same idea as different metrics. A metric on a set doesn't calculate distance, it defines distance. And two different metrics will give you two different distances.

If you walk the distance - does your experience depend on which metric you applied? i.e. what is the empirical experience of the metric?

There will be cases where the different metrics change the result of an experiment - what does that mean?

Look - I get it. We are approaching this from different world-views ... it is entirely a philosophical difference here.

 

[WannabeNewton]

Chickens and eggs.

Haha, no not really: http://math.stackexchange.com/questions/184979/relation-between-integral-and-summation

http://math.stackexchange.com/questions/251430/about-integrals-w-r-t-the-counting-measure

 

[atyy]

Haha, no not really: http://math.stackexchange.com/questions/184979/relation-between-integral-and-summation

http://math.stackexchange.com/questions/251430/about-integrals-w-r-t-the-counting-measure

Is there such a thing as a sum of an unmeasurable set, and would that constitute an example where the sum is not a special instance of integration?

 

[Simon Bridge]

I think that shows that there exist sums where integration is not a useful strategy...

Just like there are sums where multiplication is not a useful strategy.

 

[Jorriss]

Is there such a thing as a sum of an unmeasurable set, and would that constitute an example where the sum is not a special instance of integration?

Not with respect to the counting measure. The measurable sets are the power set.

 

[atyy]

Not with respect to the counting measure. The measurable sets are the power set.

Is the notion of algebraic sum used here also a sort of integration?

http://www.math.wvu.edu/~kcies/Other/ElectronicReprints/85.pdf

 

[chiro]

I agree with Simon here: integration is just a way to add things up.

The measure dictates how its done and you need to have conditions like the one mathwonk posted, but in the end its just a fancy way of "measuring" something by adding things up.

What that measure relates to in terms of some other quantity - particularly if it has a physical meaning is something else, but mathematics doesn't care about that when it comes to abstract measures.

To understand what the integral means you just have to understand what the measure represents and what its telling you in the context of what is being integrated (and with respect to what measure).