Integral with respect to a restricted measure

Question

Let $\nu=h^n\lambda$ with $\lambda$ the lebesgue measure on $\mathbb R^n$ restricted to the square $(0,h^{-1})$ and $h \in \mathbb N$ show that if $f$ is integrable with respect to $\nu$ then $$\int_{\mathbb{R}^n} f d \nu = h^n \int_{(0,h^{-1})^n} f dx$$ First for a indicator function $\chi_A$ with $A- \nu$-measurable we have $$\int_{\mathbb R^n} \chi_A d \nu=\nu(A)=h^n \lambda(A \cap (0,h^{-1})^n)=h^n \int_{\mathbb{R^n}} \chi_{A\cap (0,h^{-1})^n}(x) dx $$ $$=h^n\int_{\mathbb{R}^n} \chi_{A}(x) \chi_{(0,h^{-1})^n}(x)dx=h^n \int_{(0,h^{-1})^n}\chi_{A}(x)dx$$ and then i try to extend to any tipe of function, is these correct? any hint or help i will be very grateful