For any non-negative Borel measurable function $f : \Bbb R^2 \longrightarrow \Bbb R,$ $$\displaystyle {\int} f(\textbf {xt})\ d\lambda_{\Bbb R^2} (\textbf {t}) = |xy| \displaystyle {\int} f(\textbf {t})\ d\lambda_{\Bbb R^2} (\textbf {t}),$$ where for $\textbf {x} = (x,y)$ and $\textbf {t} = (r,s),$ $\textbf {xt} : = (xr,ys).$
Let us first assume that both the components of $\textbf {x}$ i.e. both $x$ and $y$ be non-zero. If we can prove the above result for indicator function of Borel measurable subsets of $\Bbb R^2$ then we are through. Because then by linearity of Lebesgue integral and by Monotone convergence theorem the result continues to hold for any non-negative Borel measurable function.
Let $A \in \mathcal B_{\Bbb R^2}.$ Let $f = \chi_A.$ Then $\textbf {xt} \in A \iff \textbf {t} \in \left (\frac 1 x, \frac 1 y \right ) A.$ Since $A \in \mathcal B_{\Bbb R^2},$ it follows that $\left (\frac 1 x, \frac 1y \right ) A \in \mathcal B_{\Bbb R^2}$ and $\lambda_{\Bbb R^2} \left (\left (\frac 1 x, \frac 1y \right ) A \right ) = \frac {\lambda_{\Bbb R^2} (A)} {|xy|}.$ So we have $\chi_A (\textbf {xt}) = \chi_{\left (\frac 1 x, \frac 1 y \right ) A} (\textbf {t}),$ for all $\textbf {t} \in \Bbb R^2.$ Hence we have $$\displaystyle {\int} \chi_A(\textbf {xt})\ d\lambda_{\Bbb R^2} (\textbf {t}) = \displaystyle {\int} \chi_{\left (\frac 1 x, \frac 1 y \right ) A} (\textbf {t})\ d\lambda_{\Bbb R^2} (\textbf {t}) = \lambda_{\Bbb R^2} \left (\left (\frac 1 x, \frac 1y \right ) A \right ) = \frac {\lambda_{\Bbb R^2} (A)} {|xy|}.$$ On the other hand $$\displaystyle {\int} \chi_A (\textbf {t})\ d\lambda_{\Bbb R^2} (\textbf {t}) = \lambda_{\Bbb R^2} (A).$$ So we have $$\displaystyle {\int} \chi_A (\textbf {xt})\ d\lambda_{\Bbb R^2} (\textbf {t}) = \frac {1} {|xy|} \displaystyle {\int} \chi_A (\textbf {t})\ d\lambda_{\Bbb R^2} (\textbf {t}).$$ Which is not matching with the equation I intended to prove. Where have I made mistake? Also how do I tackle the case where both the components of $\textbf {x}$ are not simultaneously non-zero? Any help in this regard will be highly appreciated.
Thanks in advance.