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I wish to define the length of a subset of $\mathbb R^2$ in a way such that any subset representing a curve has the length of the curve.

My proposed definition is:

Let a circleset for a curve be a set of non-intersecting open circles such that the center and at least one point on the perimeter of the circle is in the curve.

The length of a circleset is the sum of the diameters of the circles in that circleset.

For a curve, let $S_r$ be the set of circlesets for that curve, where no circle has a radius larger than $r$. Define the length of the curve as the limit of the supremum of the lengths in the set $S_r$, as $r\to0$.

I've been able to show that this definition works for a few curves, and unable to find a curve where it doesn't.

For a straight line, the diameter of the circle can only be longer than the total length of the curve inside the circle, if the circle contains an endpoint of the line. At most two circles contain an endpoint of the line, so the length of a circle set is at most $2r+l$ where $l$ is the length of the line. Since it's easy to construct a circleset that has exactly the length $l$, the upper and lower limits for the supremum are $l$ and $2r+l$, which squeezes to $l$ when taking the limit as $r\to0$.

For a curve consisting of finitely many connected line segments something similar should work, though the argument that there is more curve than diameter inside a circle without the endpoint is slightly more involved for the case where the curve bends inside the circle. The lower limit of the maxima can be constructed by placing a circle on each bend.

I can't tell if it works if the curve intersects itself, and if there are infinitely many line segments.

We should also check if it works for a closed curve. For a circle there is obviously always more curve than diameter, and for any $r$ the set $S_r$ contains a sequence of circlesets where the lengths are the perimeters of regular polygons converging to the circle, so every supremum is equal to the perimeter of the circle.

The question is

Does this definition of length of subset of $\mathbb R^2$ satisfy these properties?

  1. Is the definition correct for all sets of (possibly infinitely many) line segments?
  2. If a sequence of curves converges to some curve, does the sequence of their lengths converge to the length of that curve?
  3. Is the definition correct for all curves, and if no, for what curves does it fail?
  4. If $L$ is the function that assigns length, is $$L(A \cup B ) + L(A \cap B ) = L(A) + L(B)$$

Furthermore, do we need the requirement that there is a point on the perimeter of the circle in the curve?

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    Wow. Loved every bit. But, i am not at a level where i can help u. Good luck. – 2017-01-30
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    "The length of a circleset is the sum of the diameters of the circles". Which circles? All possible circles in any circle set? What if you have an infinite number of circles in your set - why can't that sum be infinite? – 2017-01-30
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    @Paul A circleset is some set of circles with some constraints on the circles. The circles we're taking the diameter of are the circles in the set. – 2017-01-30
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    @Paul The sum might be infinite, but if the definition works, it should only be infinite for infinitely long curves. – 2017-01-30
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    Take the line segment of length 1. Put circles of radius $\frac{1}{n}$ at $\frac{1}{n}$, $\frac{2}{n}$, ...,$\frac{n-1}{n}$ then this appears to be a circle set per your definition but the sum of the diameters is $2 \frac{n-1}{n}$ with limiting value 2? – 2017-01-30
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    Shouldn't your 4th requirement be: $$L(A \cup B ) - L(A \cap B ) = L(A) + L(B)$$? – 2017-01-30
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    @Paul The distance between the circles at $1/n$ and $2/n$ is $1/n$, but since they both have radius $1/n$ they are intersecting. – 2017-01-30
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    Sorry yes, missed that bit! – 2017-01-30
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    @dimpol Let $A = B$, then $L(A\cup B) = L(A\cap B) = L(A) = L(B)$. – 2017-01-30
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    Ah my mistake. A suggestion for an edge case: try fractels. Im curious what the length of the koch-curve is according to this measure' – 2017-01-30
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    @dimpol As for the koch-curve, the corners in any iteration are also in the final shape. For each iteration of the koch snowflake we can make a circleset with circles on the corners, and I think the circles won't intersect if we make the radiuses large enough that they give the diameter of that iteration of the koch snowflake, so it should have infinite length. – 2017-01-30
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    Just a thought, but if the centre and one circumference point are on the curve, why not just use the line segment with each endpoint on the curve? – 2017-01-30
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    @Paul was that for the koch-curve? The midpoints in each iteration are not in the final koch-curve. – 2017-01-30

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