Ron wrote:
Maybe I should restate my question. Assume a receiving antenna is in the
center of a sphere and the received signal is coming in equal amounts
from all points on the surface of the sphere. Which receiving antenna
would capture more power, an omni or a high gain beam? There are no
noise and no losses.

They'll intercept equal amounts, assuming both are lossless. The
directional antenna will intercept a larger fraction than the isotropic
antenna in the directions it favors, and less in others. The total will
be be the same.

In reverse, this is equivalent to calculating the average gain of the
antennas, which is the same for all lossless antennas.

Roy Lewallen, W7EL