Let $S=\{1,2,...m\}$
And $A_{1},...,A_m\subseteq X$ be some sets
$\mu:\mathcal{P}(X)\rightarrow \mathbb{C}$
Let also
$$F(S):=\sum_{I\subseteq S}(-1)^{|I|-1}\mu(\bigcap_{i\in I}A_{i})$$
If function $\mu$ is a measure then: $$F(S)=\mu(\bigcup_{i\in S}A_i)$$
This is inclusion-exclusion formula.
But what if we consider another set $Z\subseteq S$ and following expressions:
$$H(S, Z) = \sum_{I\subseteq S-Z}(-1)^{|I|-1}\mu(\bigcap_{i\in I\cup Z}A_{i}) = \sum_{I:Z\subseteq I \subseteq S}(-1)^{|I|-1}\mu(\bigcap_{i\in I}A_{i})$$
$$G(S, Z) = \sum_{I:Z\subseteq I \subseteq S}\mu(\bigcap_{i\in I}A_{i})$$
Find formulas for $G, H$
Thank you in advance.
EDIT:
Generally i am intrested in formula for:
$$\sum_{Z\subseteq I\subseteq S}\prod_{T\subseteq I}f(T)$$ where all the $T$ have fixed size $|T|=a$ and $|Z|+1\le a$
Particularly i would like to calculate: $$\sum_{\{x\}\subseteq I\subseteq S}\prod_{\{a_1,a_2\}\subseteq I}f(\{a_1,a_2\})$$