Can we assume that "any two events" are within the same sample space? I had a question regarding the properties of probability.
If we were to say that we have "any two events $A$ and $B$," is it okay to assume that they both belong to the same sample space $S$ (i.e. $A \subseteq S$ and $B \subseteq S$)?
I'll give a specific example exercise that prompted me to ask this question:
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"Show that for any events $A$ and $B$, $\ $ $P(A) + P(B) - 1 \le P(A \cap B)$."
What I did is move the 1 over, so now we have
$$P(A) + P(B) \le 1 + P(A \cap B)$$
which is true if we assume that $P(A) + P(B) \le P(S) = 1$.
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I hope my question makes sense. Thank you for the feedback!

In this case it's implicit in the notation.
Recall the definition.  A probability space is a triple: $ \left(S, \mathcal F, \mathbb P\right)$ where $\mathcal F$ is a set of "events" (Subsets of $S$) and $\mathbb P$ is a function from $\mathcal F$ to $\left[ 0,1 \right]$.
So the expression $$\mathbb P(A) + \mathbb P(B) - 1 \leq \mathbb P(A\cap B)$$ only makes sense if $A, B \in \mathcal F$  The function $\mathbb P$ carries its domain around with it, meaning that it can only ever refer to one sample space.
In addition the term event is ambiguous if there are two sample spaces involved. So the use of the term event usually means that there is only one sample space under consideration. Which means that "Any two events $A$ and $B$" can be taken to imply that $A$ and $B$ are events in the same sample space.
