If $f$ is a Lebesgue measurable function on $\mathbb{R^n}$ and for a linear map $T$ such that $T \notin GL_n(\mathbb{R})$ is $f \circ T$ Lebesgue measurable? Also what if $f$ is Borel measurable function?
I do know $T$ is continuous since it is a linear map. I'm confused because since $T \notin GL_n(\mathbb{R})$ we cannot write $(f \circ T)^{-1} = T^{-1} \circ f^{-1}$ and thus can't conclude if it takes a Borel/Lebesgue measurable set back to a Borel/Lebesgue measurable set.