Let $X$ be a sample space. and $p$ be a pdf(pmf). Let supp(p)=$\{x:p(x)>0\}$.Letting $X$ be redefined as supp(p), is equivalent to assuming that $p(x;\xi)>0$ holds for all $\xi\in E$ and for all $x\in X$.This means that $S$ is a subset of $$P(X)=\{p:X\to\mathbb{R}:p(x)>0(\forall x \in X),\int p(x)dx=1\}$$
The Coefficient of the Fisher information matrix is defined $$g_{ij}(\xi)=E_{\xi}[\partial_{i}l_{\xi}\partial_{j}l_{\xi}].............(1)$$ where, $E_{\xi}$ denotes the expectation with respect to the distribution $p_{\xi}.$ It is also possible to write $g_{ij}(\xi)$ as $$g_{ij}(\xi)=-E_{\xi}[\partial_{i}\partial_{j}l_{\xi}]...............(2)$$ and we also have $E_{\xi}[\partial_{i}l_{\xi}]=0.............(3)$
Now lets us define $S=\{p_{\xi}/\xi\in E\}$ as a subset of $$\bar{P}(X)=\{p:X\to\mathbb{R}/p(x)>0 \forall x\in X,\int p(x)dx<\infty\}$$
Then in this case the Fisher metric is the same as defined in equation $(1)$, But equation $(2)$ and $(3)$ does not hold. I didn't get why $(2)$ and $(3)$ does not hold for this case.Someone please explain. Thanks
Expression 3 needs the following to be true: $$ 0 = \frac{\partial}{\partial \zeta_i} \int_X p_x(x,\zeta) dx = \int_X \frac{\partial}{\partial \zeta_i} p_x(x,\zeta) dx$$
Expression 2 needs the following to be true: $$ 0 = \frac{\partial}{\partial \zeta_i} \frac{\partial}{\partial \zeta_j} \int_X p_x(x,\zeta) dx = \int_X \frac{\partial}{\partial \zeta_i} \frac{\partial}{\partial \zeta_j} p_x(x,\zeta) dx$$
This swapping makes both these integrals evaluate to $0$ since $\int_X p_X(x,\zeta) dx = 1$ in the original set, making the derivative $0$ and make equation/expression 2,3 to be true.
This swapping of integral and derivative need not be true in general in the set of $p_X(x,\zeta)$ you are defining as new set. Even if the swapping of integral and derivative is true, we need not have $\int_X p_X(x,\zeta) dx = constant$ to have the derivatives of integrals become $0$. So the above derivatives of integrals are not $0$. So expression 2,3 are not true.
Check when can we interchange integration and differentiation for information on when u can interchange derivative and integrals.