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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 $\infty$)?

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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
2  
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

1 Answer 1

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

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

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