  We consider the Bose-Einstein condensation (BEC) in a relativistic pion gas.
The thermodynamic limit when the system volume $V$ goes to infinity as well as
the role of finite size effects are studied. At $V\to \infty$ the scaled
variance for particle number fluctuations, $\omega=< \Delta N^2>/< N>$,
converges to finite values in the normal phase above the BEC temperature,
$T>T_C$. It diverges as $\omega \propto V^{1/3}$ at the BEC line $T=T_C$, and
$\omega \propto V$ at $T<T_C$ in a phase with the BE condensate. Possible
experimental signals of the pion BEC in finite systems created in high energy
proton-proton collisions are discussed.
