Ex4-4

Question: Let f be defined and measurable in \mathbb{R}^n. If T is a nonsingular linear transformation of \mathbb{R}^n, show that f(T\mathbb{x}) is measurable.
Solution: 
Let E_1 = \{ x, f(x)>a \} and E_2 = \{ x, f(Tx)>a \}.
So, T^{-1}E_1 = \{ T^{-1}x, f(x)>a \} = \{ T^{-1}x, f(TT^{-1}x)>a \} = \{ y, f(Ty)>a \} = E_2
 by let y = T^{-1}x.
Claim that linear transformation is a Lipschitz transformation. The proof of this statement can be found here.
By theorem, 3.33, E_2
 is measurable.
Hence, f(T\mathbb{x})
 is measurable.