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Sets of Measure Zero

∞ points can have zero area

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Sets of Measure Zero

011Total covering area: 0.007500

Set

Countably many points in [0,1]²

ε0.500
Number of points20

Covering

Rectangles20
Total area0.007500
Measure= 0
Measure → 0 as ε → 0

For n-th point: size = ε/2ⁿ

Σ ε/2ⁿ = 2ε → 0

💡 Why infinitely many points = zero area

Cover each point with a rectangle of size ε/2ⁿ. Total area = Σ ε²/4ⁿ = ε²·4/3. As ε → 0, area → 0. Countably many points occupy zero area, even though there are infinitely many.