## Inverse mean value property of harmonic functions.(English)Zbl 0794.31001

Let $$A$$ be a subset of $$\mathbb{R}^ d$$, $$d\geq 2$$, having finite Lebesgue measure $$\lambda(A)$$ and let $$B$$ denote the open ball of centre 0 such that $$\lambda(B)= \lambda(A)$$. The following result is the basis for similar results on various classes of harmonic functions: The equality $h(0)= \lambda(A)^{-1} \int_ A h d\lambda$ holds for every bounded function $$h=G^{1_ C \lambda}- G^{1_ D \lambda}$$, $$C$$, $$D$$ compact subsets of $$\complement A$$, if and only if $$\lambda(B \setminus A)=0$$.

### MSC:

 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 26B15 Integration of real functions of several variables: length, area, volume

### Keywords:

finite Lebesgue measure; bounded function; compact subsets
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### References:

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