Show that $\mu (\left[a, b\right)) = \ln \frac{1 + b}{1+a}$ (a measure on $X = [0,1]$) is invariant wrt. $x \rightarrow \left\{\frac{1}{x}\right\}$

Question

Show that $\mu (\left[a, b\right)) = \ln \frac{1 + b}{1+a}$ (a measure on $X = [0,1]$) is invariant wrt. $x \rightarrow \left\{\frac{1}{x}\right\}$

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My question here is: What does it mean, that a measure is invariant with respect to a map/function? And how can we show it here on this example, especially as we're dealing with a mapping which goes from $x \rightarrow \left\{\frac{1}{x}\right\}$ (where $\left\{\frac{1}{x}\right\}$ is the mantissa of $\frac{1}{x}$).