Mergelyan-Wesler Theorem

Theorem

Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero.

Let $r_n$ be the radius of $D_n$.

Then:

$\ds \sum_{k \mathop = 1}^\infty r_k = +\infty$

Proof

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Source of Name

This entry was named for Sergey Nikitovich Mergelyan and Oscar Wesler.

Sources

• 1962: S.N. Mergelyan: Uniform approximations to functions of a complex variable (Amer. Math. Soc. Transl. Vol. 3: pp. 294 – 391)
• 1960: Oscar Wesler: An infinite packing theorem for spheres (Proc. Amer. Math. Soc. Vol. 11: pp. 324 – 326)  www.jstor.org/stable/2032977
• 1966: Z.A. Melzak: Infinite packings of disks (Canad. J. Math. Vol. 18: pp. 838 – 852)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,30695 1 \ldots$