[dinkelberg]

Why should it not allow for your spiralling brushstroke? You can divide the spiral into the strokes corresponding to each turn around the center. We can assume that each of these strokes is measurable (otherwise we would not need the construction of the spiral). Then the entire spiral is just a countable union of those measurable strokes, thus it is measurable itself.

[Y_Y]

For the proof given in TFA they use the assumption that the symbol is compact and hence closed. I argue that the symbol needn't be closed, it may still be measurable.