Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have trouble with understanding following from my text book in Measures and Integral theory.

Let T be an orthogonal $n\times n$ matrix. If $\lambda^{n}$ is the Lebesgue measure then we have: $\lambda^{n} = T(\lambda^{n})$

And my question is:

How to interpret the $T(\lambda^{n})$-part?

How is it possible to multiply a $n\times n$ matrix with a real number ($\lambda^{n}$ is a measure and therefore it is a value between 0 and $\infty$)?

share|cite|improve this question
A measure is not a number. A measure associates numerical values to measurable sets and those values range between $0$ and $\infty$. I am however also puzzled by the notation. I suppose $\lambda^n$ means the Lebesgue measure on $\mathbb{R}^n$ and the statement means that this measure is invariant under orthogonal transformations. – Raskolnikov Sep 14 '12 at 17:01
$\lambda^{n}$ is the Lebesgue measure on $\mathbb{R^{n}}$. I know a measure is a map. But how to take a $n\times n$ matrix T to a map which is not a $n$ vector, but which output is a real number? – Guestfromthepast Sep 14 '12 at 17:05
Actually $T(\lambda^n)$ is a notation for $\lambda^n\circ T$. Which means you apply $T$ to whatever measurable set you feed to your measure and then compute the measure. – Raskolnikov Sep 14 '12 at 17:09
Ahhh.. That makes sense! Thanks! :D – Guestfromthepast Sep 14 '12 at 17:13
Is this your book? – leo Sep 15 '12 at 17:35
up vote 3 down vote accepted

$T(\lambda^n)$ is just notation for $\lambda^n \circ T$, i.e. apply the transformation T on the set before calculating its measure. So, what one asks to prove is for any measurable set $B \subset \mathbb{R}^n$ you have to show that

$$\lambda^n(B)=T(\lambda^n)(B)\equiv\lambda^n\circ T(B)\equiv\lambda^n(T(B)) \; .$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.