Ville's inequality is the theorem that makes 'valid no matter how often you look' true rather than wishful.
Also known as: Ville's maximal inequality
Ville's inequality is the theorem that a non-negative process that does not grow in expectation — a martingale starting at one — has only an α chance of ever reaching the level 1/α, across all time at once. Not at one chosen moment: ever. It is the quiet engine under everything anytime-valid.
That single guarantee is what lets you check a monitor as often as you like. If you raise the alarm when the evidence crosses 1/α, the theorem caps your false-alarm rate at α no matter how many times you peeked along the way. The peeking problem simply does not apply, because the bound was proven over every look simultaneously.
In practice you never compute Ville's inequality by hand; it is the reason the e-process threshold is what it is. When ValidAnytime says an alarm carries a guarantee, this is the theorem the guarantee traces back to.
ValidAnytime turns these ideas into a live alarm you can trust — valid no matter how often you look. Prove it on your own data, free.