Either space is finite or it is infinite.
a) - If space is infinite in extent, either it is thermal over an infinite volume, or it is in the vacuum state for most of it. If it is thermal, infinity is a large place, and random statistical fluctuations alone will ensure there are infinitely many places in the universe just like the Earth (or maybe just your room, or your brain), but not necessarily beyond. This is Max Tegmark's level I multiverse. Only a doubly exponentially small fraction of the universe will be like that. Doubly because human length scales are already exponential compared to the microscopic scale, so it's exponential of exponential. But infinity is a large place. If most of the spatial extent of the universe is in the vacuum state, in quantum field theory, we have the Reeh-Schlieder theorem for any bounded region, and so, there is still a doubly exponentially small probability that the region will be just like your brain.
b) - If space is finite: either it remains bounded in size over all time, or it expands indefinitely, or it will end in a big crunch.
b1) -If it is finite and bounded, it will undergo Poincare recurrences, and there will be infinitely many brains just like yours spread out over time.
b2) - If it expands without bound, eventually the volume of space will exceed doubly exponential, and there will be many replicas of your brain spread out over space in the future.
So why aren't we Boltzmann brains? In general relativity, two locally identical Boltzmann brains can be related by a diffeomorphism. There is no absolute position or absolute time distinguishing them. So we have to identify all these Boltzmann brain replicas relationally. Are they all the same brain, all those infinitely many of them? Infinitely many identical brains, all identified. Quantum evolution is nondeterministic, so they don't all evolve the same. So figure out the fraction of identical brains evolving in a certain manner, and we get frequentist probabilities. This requires a measure over these brains. There is no invariant measure in eternal inflation, the measure problem.