I am thinking about this question but I can not solve. The question is: Can I define Lebesgue measure in the space of parametric functions and if the answer is yes what is that Lebesgue measure? Could anyone please help me to solve this doubts??
Here is the set up:
Let us consider the space of parametric functions $f:[0,1]\rightarrow \mathbb{R^2}$. For example, $f(t)=(x(t),y(t))'$, where, $x(t)=\sin t$ and $y(t)=\cos t$.
We can define the arc length of $f(t)=\int_0^1\sqrt{\big(x'(t)^2)+y'(t)^2\big)}dt$.
Then how to address the first to question?? Thanks in advance.