  We study the Lelong classes $\mathcal{L}(V),\mathcal{L}^+(V)$ of psh
functions on an affine variety $V$. We compute the Monge-Amp\`ere mass of these
functions, which we use to define the degree of a polynomial on $V$ in terms of
pluripotential theory (the Lelong degree). We compute the Lelong degree
explicitly in a specific example. Finally, we derive an affine version of
B\'ezout's theorem.
