# Tagged Questions

128 views

### Absolute continuity of probabilities on Polish spaces and open sets.

On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
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### Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions ?

Does anyone knows whether the set of the absolutely functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the Banach ...
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### Convergence of probability measure and the *-weak convergence ?

Given a Polish space X, I note $C_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and and $(C_b(X))^\star$ its topological dual with the $*-$weak ...
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### Compactness of sigma-algebra for the $L^1$ metrics

Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metrics $d(A,B)=μ(AΔB)$. Is $(G,d)$ a compact metric space? When $F$ is ...
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### Injective with Finite Discontinuities Mapping from $\mathbb R^n$ to $[0,1]$

Hi, as a continuation to the fully answered question: Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$ Can one think of an injective $f:\mathbb R^n\rightarrow[0,1]$ that has only ...
### Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$
Hi, Can one think of an injective and Riemann integrable $f:\mathbb R^3\rightarrow\mathbb R$? (of course it cannot be continuous)