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Let $rdr\times d\theta$ be a measure on Borel subsets of $[0,\infty)\times[0,2\pi)$ and $T$ is a mapping such as $T:[0,\infty)\times[0,2\pi) \rightarrow \mathbb R^2, \ T(r, \ \theta) = (r\cos\theta, \ r\sin\theta)$.

Show that pushforward measure $T(rdr\times d\theta)$ is a two-dimensional Lebesgue measure $\lambda$ on $\mathbb R^2$.

As I suppose we have to find $(rdr\times d\theta)(T^{-1}(E))$, where $E$ is an elementary set of $\mathbb R^2$, for example a rectangle. Then conclude that pushforward measure coincide with $\lambda^2$ on $\mathbb R^2$.

Is it correct so far? Do I have to find $T^{-1}(x,y)$?

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    that should work, I think another way would be to prove that it is translation invariant and hence a multiple of Lebesgue measure. – 2017-01-31
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    Could you help me to finish my way? I mean this whole thing is getting a little bit blurry. First we have to show that $T$ is bijective (to get $T^{-1}$), then I have $T^{-1}(x,y) = (\sqrt{x^2+y^2}, arctg(\frac{y}{x}))$. But then I want to find pushforward measure such $(rdr\times d \theta)(T^{-1}([0,1]\times [0,1])$. It means that $0 \le r \le \sqrt{2}$ and $0 \le \theta \le \frac{\pi}{2}$ and how do we count that $rdr \times d\theta$? – 2017-01-31

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By definition of the pushforward measure,

$T_*(rdr\times d\theta)(E)=(rdr\times d\theta)(T^{-1}(E))$ for any measurable set $E$.

Then, since $T$ is a diffeomorphism, if $E$ is a basic rectangle,

$\lambda (T\circ T^{-1}(E))=\lambda (E)=\int_{T^{-1}(E)}|\det D_{(r,\theta) }T|d\lambda =\int_{T^{-1}(E)}rdrd\theta =(rdr\times d\theta)(T^{-1}(E))=T_*(rdr\times d\theta)(E).$

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    I forgot to say that Jacobi's transformation is not available. – 2017-01-31
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    At some point you will have to use the fact that $T$ is a diffeomorphism, to relate the two measures. Start with a basic rectangle, and use the usual calc III arguments to get the formula from scratch. I do not think there is an easier way.Or maybe I am making it too hard? Or maybe show the measure is translation-invariant would be another way? – 2017-01-31
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    You know what, I know how to show it theoretically. I will just learn Jacobi's transformation by myself :) – 2017-01-31