We are given that $X \sim Unif(a,b)$ and $u(X)$ is a statistic.

I do not understand why 

$\int\_a^b u(x) dx=0$ then $Pr[u(X)=0]=1$

What I can conclude is that 

$$\int\_0^a u(x)dx = \int\_0^b u(x)dx$$
but I am not sure how that connects to the probability expression on the right side.

What bothers me most is the fact that the probability is not an interval so I do not know how the probability is not actually $0$.

$Edit$

Thank you for your help.

It seems like the information I have provided was not sufficient so I will show everything that I was given.

$X \sim Unif(a,b)$, $u(X)$ is an unbiased and sufficient statistic.

My goal is to prove that $X$ is a complete distribution, so I must show

$$E[u(X)]=0 \quad \text{then} \quad u(X) =0$$

This leads to the equation I showed above, but I am not sure how it connects to the probability statement.

I also considered showing that $X$ is in the regular exponential class but since the support depends on $a$ and $b$ I could not do that.