Calculating average over a function set Non-math version of the problem: I am running with a GPS device, recording my path. I know the curve the GPS has recorded. However, the GPS device actually has an accuracy, which I can assume to be fixed (for starters). Assuming that the curve the GPS device records is continuous and that the actual curve is also continuous (as I do not possess the power of teleportation :) how can I calculate the expected length of my run? Now, I can translate the problem in math terms: I am looking for the average length of a function $f:[0,T]\mapsto\mathbf{R}^2$, such that $f \in C^1[0,T]$ and $\|f-g\|_{\infty}<c$, where $g \in C^1[0,T]$ is a fixed function (the recorded data), $c$ is a constant (the accuracy of the GPS) and $T$ is the time it took me to run the distance. However, when it comes to the solution, I have absolutely no idea. I have not studied much functional analysis, so if this is a trivial exercise, sorry. Any pointers would be appreciated.

If you were drunk your path may look like $f(t)=g(t)+c\sin(\omega t)$ for sufficiently large $\omega $. In this case your path $f$ will not significantly differ from $g$ in $\Vert\cdot\Vert_\infty$ metric, but the length of your path could be arrbitrary large. Hence you need to reformulate your problem in terms of $\Vert\cdot\Vert_1$ metric or put some restrictions on possible trajectories.