dumbed down explanation of Fubini's theorem? [duplicate]

Question

Could someone please give me a simple explanation of when it is valid to interchange the order of Intergration and summation when the summation is an infante summation?

for some context during lecture the other day my current prof interchanged the order, and I asked if that were valid here and he said he couldn't think of any reason it wouldn't be so I said that I remembered a theorem about when that was valid. After class I told him i thought it was fubini's theorem and that the "function" inside the summation had to be Lebesgue integrable and he didn't know what that meant.

if someone could try to help me understand this and to explain the concept to him that would be very helpful.

Answer 1

As you have correctly noted, exchanging a series and an integral (or two series/integrals) is a nontrivial thing because you have to exchange two limiting processes which in general leads to different results.

The general form of the Fubini-Tonelli theorem gives a handy sufficient condition when it is possible to do exchange the order of integration: If a function is absolutely integrable in one specific order of integration, then it is automatically integrable in any order of integration and all orders yield the same result, i.e. you can exchange the integration processes as you like. Formally, if a measurable function $f: \mathbb{R}^n \to \mathbb{R}$ satisfies

$$ \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \lvert f(x_1, \dots, x_n) \rvert dx_1 \cdots x_n < \infty, $$

then

$$ \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(x_1, \dots, x_n) dx_{\sigma(1)} \cdots x_{\sigma(n)} = \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(x_1, \dots, x_n) dx_{\rho(1)} \cdots x_{\rho(n)} $$

for any permutations $\sigma$ and $\rho$.

As series are just a special variant of Lebesgue integrals, the statement also holds for series and any combination of integrals and series.

Answer 2

The classical long way via measurable sets:

1. A set is measurable if its inner measure (supremum of union of covers by open sets from inside) and outer measure (infimum of union of covers by closed set from outside) coincide.

2. The n-dimensional integral over a measurable set in $\mathbb R^n$ is its volume.

3. The n-dimensional integral over a positive, smooth function is its $L_1$ norm wrt to that a measurable set. As the limit of series of positive, bounded terms, its independent of the division of the volume into a complete cover of nonoverlapping, measurable subsets and the order of summation.

4. A positive function is called (Lebesgue) measurable, if it is the limit of step functions on a subdivision of the measurable volume in measurable subdivisions.

5. The Lebesgue integral is defined as the limit of sums with order according to the value of the functions on the subdivisions.

6. The Lebesgue integral over a measurable set $V\subset \mathbb R$ over a measurable function is $$\int_V f(x) dx := \int_{-\infty}^\infty y \ \mu \left( f^{-1}(dy) \right)$$ with $\mu$ the Lebesgue measure of measurable sets in $\mathbb R$, $f^{-1}$ the inverse measurable function as a map between sets and $dy$ a symbol for limits of the division of $\mathbb R$ into translation invariant open intervals.

7. For positive measurable function interchange of limits leaves integrals invariant.

8. For nonpositive functions the same is true if the positive and absolute of the negative parts are measurable.

9. Riemann integrable functions are measurable, integrals coincide.

10. (The hard part) The construction of a non measurable function implies the use of transfinite methods, Zorns lemma, axiom choice, meaning, at a certain point one takes a limit, that cannot be discribed as a formula or an algorithm. So physicists e.g. may interchange limits and the order of integration as long as measurability of sets and function are guaranteed.

Beware of improper integrals and distributions.