4. The number of elements in the set $\left\{(x, y) \left\lvert\, \lg \left(x^{3}+\frac{1}{3} y^{3}+\frac{1}{9}\right)=\lg x+\lg y\right.\right\}$ is
A. 0
B. 1
C. 2
D. More than 2
By the AM-GM inequality, we have $x^{3}+\frac{1}{3} y^{3}+\frac{1}{9} \geqslant 3 \sqrt[3]{\frac{1}{27} x^{3} y^{3}}=x y$, equality holds when $x^{3}=\frac{1}{3} y^{3}=\frac{1}{9}$ $\Rightarrow x=\sqrt[3]{\frac{1}{9}}, y=\sqrt[3]{\frac{1}{3}}$. Thus, the set contains only one element, so the answer is $B$.