Suppose the state of the world as a function of time is , and the value of a state of the world is $V (x)$ . The orthodox way to aggregate the value of a future trajectory is exponential discounting, i.e., $U (X) = \int_{0}^{\infty} γ^{t} \cdot V (X (t)) d t$ , where $0 < γ < 1$ is called the discount factor. Now, in your example problem, let us take $V (status quo) := 0$ , $V (1 life saved) := 1$ , and $V (7 billion tortured) = - (7 \times 10^{9}) ε$ , where I'll let you choose $ε$ , but it has to be strictly greater than zero. (Assuming you don't think the torture itself is neutral, your value function should have this form, up to additive and multiplicative constants.)

In the status quo,

U (status quo) = \int_{0}^{\infty} γ^{t} \cdot V (status quo) d t = 0

Pressing the button gives

\begin{matrix} U (press button) = & \int_{0}^{10^{11}} γ^{t} \cdot V (7 billion tortured) d t + \int_{10^{11}}^{\infty} γ^{t} \cdot V (1 life saved) d t = & \frac{1}{- log γ} [(1 - γ^{10^{11}}) V (7 billion tortured) + γ^{10^{11}} V (1 life saved)] = & \frac{1}{- log γ} [- (1 - γ^{10^{11}}) (7 \times 10^{9}) ε + γ^{10^{11}}] \end{matrix}

If we solve the inequality for this to be strictly positive, we get

γ > {(\frac{(7 \times 10^{9}) ε}{1 + (7 \times 10^{9}) ε})}^{10^{- 11}}

Supposing $ε$ is, I don't know, 0.001 for example, then $γ$ has to be greater than $0.999999999999999998$ . Well, you might say, isn't that still less than $1$ ? Sure, but:

$γ$ should be bounded above not just by $1$ , but by $1 - P (x-risk per year)$ . In this case, your intuition for $γ$ seems to be pushing the probability of extinction per year down to at most $10^{- 17}$ . There's an argument to be made that aligned AGI would get the extinction risk of Earth-originating life down to something like that kind of level, but it's not trivial (extraterrestrial threats start to get plausible), and you don't seem to be making it.
the large numbers involved here, and their arbitrary nature, suggest that your intuition is guiding you along the lines of $γ = 1$ , and there are all sorts of inconsistency paradoxes from that, because the expected value of a lot of things ends up being $\infty - \infty$ , which is really bad, even measure theory can't save you from that kind of divergence. I'm as long-termist as the next guy, but I think we have to set our discount factors a little lower than $0.999999999999999$ .