A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He misses (with probability one) of course, because the set of angles in which can hit the tree have measure zero, i.e. while there are an infinite number of trees, they are countably infinite.
What happens when the hunter takes many shots, whose angles make up the set $A$ when
- $|A|=c$, where c is finite
- $|A|=|\mathbb{Q}|$, a countably infinite number of shots
- $|A|=|\mathbb{R}|$, a uncountably infinite number of shots
When the hunter takes a finite number of shots as in 1. he still misses each shot with probability one. What about 2. and 3.? My intuition says that he still misses with probability one for all but the last one, as you can map each shot from 2. onto the rational points over $\mathbb{Q}^2$, but I'm not sure how to prove this.
Edit: At Qiaochu's suggestion, the text of question has been changed from always misses to misses with probability one.
If the hunter fires a gun once in a direction uniformly chosen at random with respect to the unique translation-invariant probability measure on angles, he misses with probability $1$. This is not the same as saying that he always misses; there's a non-empty measure-zero event corresponding to him hitting something. In probability theory the usual way to say this is that he misses almost surely, to emphasize that he doesn't miss surely. See also this math.SE question.
This remains true for a finite number of shots, but after that there are issues with defining infinite product measures that I'm not familiar with. I think things are okay for countable products of probability spaces but I don't know what happens after that.