Take the space of infinite coin flips where each outcome is either win (W) or loss (L) with a uniform measure. I want to know the measure/probability of the sequences that have a win-loss rate always below 1/2
Example: (LLLLL...) belongs since its win-loss rate is always 0. (LWLLLLLL...) does not belong as its win-loss rate at t=2 was exactly 1/2 (even though the sequences win-loss rate goes to 0 eventually)
I tried to write out how the sequence must start to see if there was a pattern. For example it must start with L. Let [L] be the set of sequences that start with L. It has measure 1/2. Continuing this way
Time 1: [L] measure 1/2, Time 2: [LL] measure 1/4, Time 3: [LLL], [LLW] measure 2/8, Time 4: [LLLL], [LLLW], [LLWL] measure 3/6
Edited to explain the situation better
The probability that the win-loss rate is always below $1/2$ is actually $0$. This is a standard question in probability; let $S_n$ be the number of wins minus the number of losses. Then $S_n$ is a simple random walk, which is recurrent, meaning that $$P[S_n = 0 \text{ for infinitely many }n ] = 1\,.$$
There's a nice discussion about this on Wikipedia: https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk