Suppose $g(t)$ is Gaussian random variable for real $t \in D $. Assume $[t,t+\Delta t] \subseteq D $.
I want to prove that, for \begin{equation} \xi = \int_t ^{t+\Delta t} g (\tau ) d\tau \end{equation} $\xi$ is Gaussian random variable.
I already know that for random variables $X_1, X_2$ with normal distribution, \begin{equation} X = X_1 + X_2 \end{equation} is have normal distribution. However, I have difficulty on continuous and integral instead of discontinuous and sum.
Does anyone have idea or counter examples of my question?
It is not true in general that the sum of Gaussian random variables is Gaussian - this is only true if they are independent, as well. In a similar way, your claim about the integral is not true without some further assumptions on the covariance structure of your Gaussian process $g(t)$.
A well-known counterexample goes as follows. Let $X,Y$ be independent standard Gaussians, and let $$ Z=\text{sign}(X)\cdot |Y|. $$ Then $Z$ is also standard Gausian, but it can be shown that $X+Z$ is not Gaussian at all.