Weak convergence vs Convergence in Measure What is the difference between weak convergence and convergence in measure? example let 
$\mu_{n}\Rightarrow \mu$ on $[0,1]$ (the space where all measures are defined is $[0,1]$.)
How does this contrast with the statement that we have a sample space $\Omega$ on which are defined random variables $X_{1},\ldots,$ which map $\Omega $ to $[0,1]$ and $X_{n}$ converges in measure to some random variable $X$.
 A: Weak convergence is weaker than convergence in measure. A very easy and illuminating example is to note that for weak convergence, the only requirement is that the distributions of the random variable converges. So if we pick for instance $X$ to be Gaussian (or something symmetric), and consider the sequence $X,-X,X,-X,\ldots$ (Ok, for your case you may need to pick a $[0,1]$ valued r.v. symmetric about $0.5$ and use $X,1-X,X,1-X,\ldots$). This sequence trivially converges weakly (as the distributions in the sequence are identical) but it's clear that this sequence does not converge in measure.
PS if the weak convergence is to a degenerate distribution (constant), then the convergence is also in measure, if I remember correctly.

If I interpret your remarks correctly, you are asking about the induced measures on $[0,1]$, throwing out the original sample space. Then you talk about the same notion of convergence, which is that $\mu_n(A) \to \mu(A)$ for sets $A$ without atoms of $\mu$, where these are measures on $[0,1]$. This notion relates to the weak* convergence of measures as linear functionals on the space of continuous functions. This is all in the realm of real analysis.
But then you ask about a sample space for probability and notions of convergence in measure (probability). For these, the sample space matters a lot! (As my example shows). To even talk about a sequence of random variables generically you have to use a sample space of the form $\Omega^\mathbb{N}$.
If we go back to just $[0,1]$, are you then asking how it relates to other notions of convergence? (ptwise, and in measure convergence make sense only for functions, say densities, but not for generic measures).
A: Weak convergence is the convergence of measures given the weak* topology of $ C([0,1]) $ by Riesz-Representation theorem, which implies $ \mu_n \overset{*}{\rightharpoonup} \mu $ iff for all measurable functions $f$ we have 
$$ \int_0^1 fd\mu_n \rightarrow \int_0^1 fd\mu $$
Whereas convergence in measure is convergence of measurable functions, $ f_n \overset{\mu}{\rightarrow} f $ iff as $ n\rightarrow \infty $ we have 
$$ \mu(\{x \in [0,1]\ |\ |f_n(x)-f(x)|> \epsilon \}) \rightarrow 0 $$
Thus $ X_n \overset{\mu}{\rightarrow} X $ iff $ \mathbb{P}(|X_n-X|>\epsilon) \rightarrow 0 $ as $ n \rightarrow \infty $ 
