  The measurement of positron--nucleus collisions was used to estimate the
possibility of suppressing background events that simulate $\beta\beta$ decay
in the emulsion region adjacent to molybdenum conglomerates. The range of the
escape of two relativistic particles from the interaction was found to be $<d>
= (0.60\pm 0.03) ~\mu$m, which approximately corresponds to the grain size of
developed nuclear emulsion. No correlation of the values of d with the angle
between two relativistic particles was observed. It was shown that it was
possible to exclude $\beta\beta$ decay background from electrons emerging in
the decay of elements of naturally occurring radioactive chains. The background
from $\beta$ decays of $^{90}$Sr and $^{40}$K available in emulsion around Mo
conglomerates was determined by the ratio of the volume $(\sim d^3)$ to the
total volume of emulsion and was found to be $1.5\cdot 10^{-2}$. It was shown
that the backgrounds from $^{40}$K, $^{90}$Sr and natural radioactivity could
be significantly suppressed and would not limit the sensitivity of the
experiment with 1 kg $^{100}$Mo.
