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2 I TeXed the mathematics.

Consider the probability space (Ω, B$(\Omega, {\cal B}, λ) \lambda)$ whereΩ=(0,1), B $\Omega=(0,1)$, ${\cal B}$ is the Borel sets, and λ $\lambda$ is Lebesgue measure.

For random variables W,Z $W,Z$ on this space, we define the Ky Fan metric by

α(W,Z)=

$$\alpha(W,Z) = \inf { ε>0: λ(|W-Z| ε) ε}.\lbrace \epsilon > 0: \lambda(|W-Z| \geq \epsilon) \leq \epsilon\rbrace.$$

Fix the random variable X(ω)=ω, $X(\omega)=\omega$, so the law of X $X$ is Lebesgue measure,that is, law(X)=λ.${\cal L}(X)=\lambda$.

Question: For any probability measure μ $\mu$ on ℝ, $\mathbb R$, does there exist a random variable Y $Y$ on (Ω, B$(\Omega, {\cal B}, λ) \lambda)$ with law μ $\mu$ so that α(X,Y) $\alpha(X,Y) = \inf {α(X,Z) \lbrace \alpha(X,Z) : law(Z{\cal L}(Z) = μ}?\mu\rbrace$ ?

the infimum above is dP(λ,μ):That is, there exist X1$X_1$ and Y1$Y_1$ on (Ω, B$(\Omega, {\cal B}, λ)with law(X1${\cal L}(X_1) = λ, law(Y1\lambda$,${\cal L}(Y_1) = μ, \mu$, andα(X1,Y1)$\alpha(X_1,Y_1) = dP(λ,μ).But in my problem, I want to fix the random variable X.$X$.

space L0(Ω, B$L^0(\Omega, {\cal B}, λ) \lambda)$ is huge. Thereare lots of random variables with law μ. $\mu$. I can't think of any space L0(Ω, B$L^0(\Omega, {\cal B}, λ) \lambda)$ is huge. A compactness
1

# Is the infimum of the Ky Fan metric achieved?

Consider the probability space (Ω, B, λ) where Ω=(0,1), B is the Borel sets, and λ is Lebesgue measure.

For random variables W,Z on this space, we define the Ky Fan metric by

α(W,Z)= inf{ ε>0: λ(|W-Z| ≥ ε) ≤ ε}.

Convergence in this metric coincides with convergence in probability.

Fix the random variable X(ω)=ω, so the law of X is Lebesgue measure, that is, law(X)=λ.

Question: For any probability measure μ on ℝ, does there exist a random variable Y on (Ω, B, λ) with law μ so that α(X,Y) = inf{α(X,Z) : law(Z) = μ}?

Notes:

1. By Lemma 3.2 of Cortissoz, the infimum above is dP(λ,μ): the Lévy-Prohorov distance between the two laws.

2. The infimum is achieved if we allowed to choose both random variables. That is, there exist X1 and Y1 on (Ω, B, λ) with law(X1) = λ, law(Y1) = μ, and α(X1,Y1) = dP(λ,μ). But in my problem, I want to fix the random variable X.

3. Why the result may be true: the space L0(Ω, B, λ) is huge. There are lots of random variables with law μ. I can't think of any obstruction to finding such a random variable.

4. Why the result may be false: the space L0(Ω, B, λ) is huge. A compactness argument seems hopeless to me. I can't think of any construction for finding such a random variable.