I'm wondering whether we can define a uniform distribution on the space of continuous functions over a compact set, e.g. $C([0, 1])$. If so, then how should I rigorously describe it? And how can I numerically `draw' a random function from this distribution? If not, then what additional assumption/constraints should I impose to have a well-defined uniform or any kind of distribution that I can easily draw samples from?
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1$\begingroup$ Perhaps Gaussian measures (see this search also) are what you're looking for. $\endgroup$– Dave L. RenfroCommented Mar 12, 2020 at 16:40
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3$\begingroup$ The most natural sense of "uniform" would be "translation invariant", analogous to Lebesgue measure on $\mathbb{R}^n$. But no such distribution exists: there is no infinite-dimensional Lebesgue measure. $\endgroup$– Nate EldredgeCommented Mar 12, 2020 at 18:30
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$\begingroup$ @NateEldredge Thanks for the reference here! If we cannot define uniform distribution on $C([0, 1])$, then what is the most natural way to define a probability distribution on it? And more importantly, how can we numerically draw samples, i.e., randomly choose functions in $C([0, 1])$, from this distribution? $\endgroup$– mw19930312Commented Mar 14, 2020 at 15:52
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$\begingroup$ @mw19930312: The most "natural" measure that people tend to consider is Wiener measure, the law of Brownian motion. This is a huge topic in probability theory. There are lots of ways to numerically approximate a Brownian motion, e.g. a random walk with small step size. Wiener measure is in particular a Gaussian measure, as Dave L. Renfro suggested. $\endgroup$– Nate EldredgeCommented Mar 14, 2020 at 15:55
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