Extending Pre-measure into two different measures. I am trying to find an example of an algebra $\mathcal{A}$, and a pre-measure $\mu_0$ such that, you can extend $\mu_0$ in the $\sigma-$algebra generated by $\mathcal{A}$ to two different measures.
By Caratheodory's extension theorem, you must have that the trivial extension is not $\sigma - $finite.
I was thinking of doing something in $\mathbb{R}^{\mathbb{R}}$, and work with integrals (cannot use $L^p(\mathbb{R})$ because it is separable), but from that idea, all I have got, are failed attempts.
Any help is appreciated.

On $[0,1)$, consider the algebra $\mathcal{A}$ of all finite unions of half-open intervals, of the form $\bigcup_{i=1}^n [x_i, x_{i+1})$.  This generates the Borel $\sigma$-algebra. Consider the pre-measure $\mu_0$ which assigns measure $+\infty$ to every non-empty set in $\mathcal{A}$.  Then one extension of $\mu_0$ to the Borel $\sigma$-algebra is counting measure $\mu$.  But $c \mu$ is also an extension of $\mu_0$ for any $0 < c \le \infty$.
