View source for Meier theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633601.png" /> be a meromorphic function in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633602.png" />; then all points of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633603.png" /> except, possibly, for a set of the first category on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633604.png" />, are either Plessner points or Meier points. By definition, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633605.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633606.png" /> is a Plessner point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633607.png" /> if the angular cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633608.png" /> is total (i.e., coincides with the whole extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m0633609.png" />) for every angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336010.png" /> between pairs of chords through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336011.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336012.png" /> is said to be a Meier point (or to have the Meier property) if: 1) the complete [[Cluster set|cluster set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336014.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336015.png" /> is subtotal, i.e. does not coincide with the whole extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336016.png" />; and 2) the set of all limit values along arbitrary chords of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336017.png" /> drawn at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336018.png" /> is identical to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336019.png" />. The theorem was proved by K. Meier [[#References|[1]]].

Meier's theorem is the analogue, in terms of the category of a set, of the [[Plessner theorem|Plessner theorem]], which is formulated in terms of measure theory. A sharpening of Meier's theorem is given in [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Meier,   "Ueber die Randwerte der meromorphen Funktionen"  ''Math. Ann'' , '''142'''  (1961)  pp. 328–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Gavrilov,   A.N. Kanatnikov,   "Characterization of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063360/m06336020.png" /> for meromorphic functions"  ''Soviet Math. Dokl.'' , '''18''' :  2  (1977)  pp. 15–17  ''Dokl. Akad. Nauk SSSR'' , '''233''' :  1  (1977)</TD></TR></table>