# What are complete sufficient statistics?

I have some trouble understanding complete sufficient statistics?

Let $T=\Sigma x_i$ be a sufficient statistic.

If $E[g(T)]=0$ with probability 1, for some function $g$, then it is a complete sufficient statistic.

But what does this mean? I've seen examples of uniform and Bernoulli (page 6 http://amath.colorado.edu/courses/4520/2011fall/HandOuts/umvue.pdf), but it's not intuitive, I got more confused seeing the integration.

Could someone explain in a simple and intuitive way?

Intuitively, if a function of $T$ has mean value not dependent on $\theta$, that mean value is not informative about $\theta$ and we could get rid of it to obtain a sufficient statistic "simpler". If it is boundedly complete ans sufficient, no such "simplification" is possible.
A complete sufficient statistic $T(x)$ is a function of summation of x whose coefficient $Q(\theta)$, if the pdf is expressed in the form of a k-parameter exponential family, has an open set in $R_k$.