Let $\mathfrak a$ be a Lie algebra graded by the abelian semigroup $S$, then the universal enveloping algebra $U(\mathfrak a)$ of $\mathfrak a$ is $S \sqcup \{0\}$ graded. I have the following questions.

1. What is the definition of infinite fold tensor product ($U(\mathfrak a)^{\otimes \infty}$) of $U(\mathfrak a)$ and is this also $S \sqcup \{0\}$ graded?
2. If so, how to express the grade spaces of this infinite tensor product in terms of grade spaces of $U(\mathfrak a)$?
3. Is it a good notation $U(\mathfrak a)^{\otimes \infty}$?

Thank you.