Can a probability measure on $[0,∞)$ with fixed mean have an arbitrarily high variance?

Question

For $C>0$ arbitrarily large, is it possible to construct a (Borel) probability distribution $\mu$ such that

1. $\mu$ is supported on $[0,∞),$
2. $\int_0^∞ \; d\mu(x) = 1,$
3. $\int_0^∞ x \; d\mu(x) = 1,\qquad$ and
4. $\int_0^∞ x^2 \; d\mu(x) = C\qquad$?

I suspect it is possible, but my intuition is telling me the opposite—since the worst weighting

Answer 1

Consider the distribution $\frac{C-1}C\delta_0+\frac1C\delta_C$.