Prove or disprove: $(X_{(1)},X_{(n)})$ is a complete statistic for $Unif(\theta-\sigma,\theta+\sigma)$
I have reduced this problem to showing that $$\int_{x=\theta-\sigma}^{\theta+\sigma}\int_{y=x}^{\theta+\sigma}g(x,y)(y-x)^{n-2}\,dy\,dx=0\,\,\forall\,\sigma,\theta>0$$ $$\implies g\equiv 0\,\,\,\,\,\,\,\forall\,g\,\text{measureable}$$ I know that $$\int_{B}g\,d\mu=0\,\,\forall B\in B(\mathbb R)\implies g\equiv0$$ How can I proceed? I had read that $B(\mathbb R)$ can be characterized by all open intervals. (That is, If know that $\int f=\int g$ over all open intervals in $\mathbb R$, then $f\equiv g$). Is there a similar thing for $\mathbb R^2$ as well?
I think my first condition says that integral is $0$ over a specific set of right angled triangles in $\mathbb R^2$ where hypotenuse is parallel to $x=y$
Note that if I assume $g$ is continuous rather than measurable, then argument boils down to showing primitive is constant which is a lot easier.
