# Class of parametrization changes which do not affect the value of the path integral

We can define the path integral of a continuous function $G: \Bbb{R}^N \to \Bbb{R}$ on every path $\gamma:[0,1] \to \Bbb{R}^N$ for which the following makes sense $$\int_\gamma G ds = \int_0^1 G(\gamma(t))|\gamma'(t)|dt.$$

We know that path integrals are independent of the parametrization for $C^1$ paths, by the change of variables formula.

I was wondering if the following is true:

Supose that we have a piecewise $C^1$ path $\gamma:[0,1] \to \Bbb{R}^N$ with $|\gamma'|>0$, without self intersections, and a Lipschitz continuous path $\beta :[0,1]\to \Bbb{R}^N$ which maps $[0,1]$ onto $\gamma([0,1])$ bijectively (i.e. $\beta$ travels along the same path as $\gamma$ but, perhaps with less regularity).

Is it true then that $$\int_\beta G ds=\int_\gamma G ds \ ?$$

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