The Martin boundaries of equivalent sheaves

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The Martin compactification of

X

defined by a Brelot sheaf

H_{1}

satisfying proportionality is shown to be the same as for

H_{2}

if the sheaves agree outside a compact set. Minimal points coincide and hence

S_{1}^{+}

and

S_{2}^{+}

are isomorphic topological cones. Nakai’s result on the extension to

X

of a function harmonic outside a compact set is extended to Bauer’s theory. The connected components of the Martin boundary

Δ

correspond to the ends of

X

which are related to direct decomposition of the cone

H^{+}