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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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    Is [this](http://books.google.es/books/about/Measure_and_Integral.html?hl=es&id=YDkDmQ_hdmcC) your book?2012-09-15

1 Answers 1

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$T(\lambda^n)$ 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

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