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I was trying to understand what the difference is between a distribution and a function. This question was answered here.

However when I read about probability distributions on Wikipedia, the article says, that such a distribution is "the mathematical function that...". So a probability distribution IS a function, yet distributions and functions are distinct mathematical objects. So a probability distribution should not be a (Schwartz) distribution, right? And probability distribution is just a fancy name for a specific kind of function, whereas a distribution is a mathematical object, right?

timtam
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    Holy crap, that Wikipedia definition is terrible. A probability distribution is not a function. I think what Wikipedia was getting at is that probability distributions on a finite set can alternatively be represented as functions, in a very natural way. However, the uniform distribution over $[0,1]$, for example, is a probability distribution, but it's certainly not a function on $[0,1]$. – mathworker21 Aug 05 '22 at 08:47

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Any probability measure $\mu$ on $\mathbb{R}^d$ can be viewed as a Schwartz distribution on $\mathbb{R}^d$, namely a continous linear map on $\mathcal{S}({R}^d)$, namely $$T_\mu : \varphi \mapsto \int_{\mathbb{R}^d}\varphi\mathrm{d}\mu.$$ The continuity of this linear form follows from the equality $$|T_\mu(\varphi)| \le ||\varphi||_\infty.$$

Christophe Leuridan
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