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Progressive measurability and functional composition
Continuity alone can't be sufficient; since $f$ is allowed to depend on $\omega$, you need some condition to ensure that it doesn't look into the future. For example if $Y(t)$ is any process that isn't progressively measurable, you could simply set $f(x,t,\omega)=Y(t,\omega)$. Then $f$ is certainly continuous in $x$ (even constant), and no matter what $X$ is, we have $f(X(t), t) = Y(t)$ which is not progressively measurable. 
Jan 27 
comment 
Reference : Special case of Banachvalued function integration by parts
What exactly is the definition of $\partial_t u$ here? 
Jan 26 
comment 
Can all unitdistance graphs have their vertices at algebraic integers?
What are you assuming about the graph $G$? Suppose I take the graph whose vertices are all the complex numbers, with an edge between two vertices at unit distance: $E = \{(z,w) : zw = 1\}$. That seems to be unit distance by your definition (let $f$ be the identity function), but since it has uncountably many vertices we can't embed it in the algebraic numbers. 
Jan 26 
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Convergence in distribution of stochastic equation solutions
Please add a link to the Math.SE post, and likewise, edit that post to add a link to this one. 
Jan 23 
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Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?
Are you using Haar measure, or some other measure? 
Jan 23 
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Book on Convergence Concepts in Probability without Measure Theory
This site is focused only on research level questions, so textbook recommendations for undergrads are off topic. math.stackexchange.com would be a better place. But the rigorous definitions of the various modes of convergence of random variables are all in terms of measure theory, so I don't think you can really study this subject properly without understanding measure theory. I agree with @NoahS that it wouldn't be a waste of your time at all (provided you have the necessary background now); you'll just find it much easier in three years. 
Jan 22 
answered  convergence in distribution and convergence of moments 
Jan 21 
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Equivalence of Gaussian measures
@AbdelmalekAbdesselam: Yes, you're right. I got a square root backward in my head somewhere. Fixed. 
Jan 21 
revised 
Equivalence of Gaussian measures
trace class covariance 
Jan 21 
comment 
Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?
Your requirement (2) seems wrong; you are essentially conditioning on your process being inside $U$ at time $t$, but it seems what you want is to condition on it staying inside $U$ all the way up to time 1. I have to say that unfortunately, I think answering this question properly requires rewriting a book on Brownian motion. 
Jan 21 
awarded  Popular Question 
Jan 20 
comment 
Equivalence of Gaussian measures
@MartinHairer: Oops, thank you. Fixed it. 
Jan 20 
revised 
Equivalence of Gaussian measures
1 implies 2 
Jan 20 
revised 
Equivalence of Gaussian measures
tags 
Jan 20 
answered  Equivalence of Gaussian measures 
Jan 20 
comment 
Equivalence of Gaussian measures
No, I don't think this is true. For instance, $N(0,C)$ and $N(0, 2C)$ are typically mutually singular. 
Jan 19 
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Problem on convergence in probability measres
@podu: Sorry about that. For probability measures it is true. See my edit. 
Jan 19 
revised 
Problem on convergence in probability measres
added 482 characters in body 
Jan 19 
revised 
Problem on convergence in probability measres
added 74 characters in body 