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I have seen this in a few tutorial and papers on Dirichlet Process, where people refer to the probability mass function of stick-breaking-process as a random measure. From the wikipedia definition there doesn't seem to be any link between these two. I wonder what is the relationship.

Note: [I am a computer science student, so there are definitely some missing stuff in my knowledge which may seem obvious to a statistics student].

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    $\begingroup$ You were one Wikipedia click away from the connection! From the page on "random element" (linked on the page for "random measure"): "In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line". $\endgroup$
    – shadowtalker
    Jul 9, 2016 at 21:50

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The Dirichlet process can be thought of as a random variable where any particular realisation is a measure. In the case of the Dirichlet process this random measure is almost surely a discrete probability measure, and may be written as $$ \sum_i w_i \delta_{X_i}, $$ where $\{w_i\}$ are distributed according to a stick breaking process and the atom locations $X_i$ are i.i.d from some distribution.

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