Intereting Posts

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Homotopy groups of $S^2$
Can someone explain Gödel's incompleteness theorems in layman terms?
When can the maximal sigma algebra be generated by all singleton subsets?
A question concerning Borel measurability and monotone functions
What is the Conjunction Normal Form of a tautology?
Equivalent characterisations of Dedekind-finite proof
The Duality Functor in Linear Algebra
Propositional Calculus and “Lazy evaluation”?
Advice for Self-Study
Orbit space of a free, proper G-action principal bundle

- I was wondering what theorem(s)

makes possible exchanging the order

of Lebesgue integrals, for instance,

in the following example:

$$\int\nolimits_0^1 \int_0^x \quad 1 \quad dy

dx = \int_0^1 \int_y^1 \quad 1 \quad

dx dy,$$ or more generally

$$\int_0^1 \int_0^x \quad f(x,y)

\quad dy dx = \int_0^1 \int_y^1

\quad f(x,y) \quad dx dy.$$ I am not

sure if it is Fubini’s theorem because I have questions regarding it in the next part. -
In Fubini’s theorem:

- Must the set over which the

double/overall integral is taken be

a “rectangle” subset, i.e. $I_1

\times I_2$, instead of a general

subset in the product space? - Must the set over which the inner

integral is taken not depend on the

dummy variable in the outer integral?

The answers to the above two questions seem to be “must” and “must not”, based on Wikipedia and Planetmath.

- Must the set over which the

Thanks and regards!

- Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$
- Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$
- Integral of differential form and integral of measure
- Integral of Thomae's function
- Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$
- How to change variables in a surface integral without parametrizing

- Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$
- Find $\int \frac{5x^4+4x^5}{(x^5+x+1)^2}$
- How to find $\int x^{1/x}\mathrm dx$
- Integral $\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx$
- Measurable functions and compositions
- Is Riesz measure an extension of product measure?
- First uncountable ordinal
- Integral evaluation $\int_{-\infty}^{\infty}\frac{\cos (ax)}{\pi (1+x^2)}dx$
- $C_c(X)$ dense in $L_1(X)$
- Why don't all odd functions integrate to $0$ from $-\infty$ to $\infty$?

Imagine that we are trying to integrate some function $f(x,y)$ over some sort of strange-shaped (bounded, for now) region of the plane, which we can denote by $\Omega$. Then we know that this region is contained in some rectangle $R$ if we simply allow $R$ to be big enough. Then we can extend our function $f(x,y)$ over all of $R$ by setting $f \equiv 0$ on all points of $R$ not in $\Omega$.

Then the double integral over $\Omega$ is equal to the integral over $R$, if they exist. And so Fubini’s theorem applies to oddly shaped regions as well.

$$

\int _0^1\int _0^xf(x,y)dydx=\int _0^1\int _0^1\chi _{[0,x]}(y)f(x,y)dydx

$$

*Now* apply Fubini to get

$$

\int _0^1\int _0^xf(x,y)dydx=\int _0^1\int _0^1\chi _{[0,x]}(y)f(x,y)dxdy=\int _0^1\int _y^1f(x,y)dxdy,

$$

where I have used the fact that $\chi _{[0,x]}(y)=0$ unless $y\leq x\leq 1$.

Techincally speaking, you can only apply Fubini (or Tonelli) for a rectangular region. To do more general regions, you have to play around with characteristic functions as I just did (or possibly even do a change of variables) and *then* apply Fubini (or Tonelli). However, in practice, with a bit of geometric intuition, you can figure out what the bounds should be without doing this.

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