If the canonical representative of a subharmonic distribution on a connected domain attains a finite global maximum at an interior point, it is constant. The ball mean inequality forces equality almost everywhere on a small ball around the maximum. The canonical representative then equals the maximum everywhere on that ball. The maximum set is open by this argument and closed by upper semicontinuity; connectedness makes it the whole domain.

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  1. Maximum principle for subharmonic functions
  2. Subharmonic function
  3. Harmonic function
  4. Partial differential equation
  5. Analysis
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