Question

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 d_P(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) $\alpha(X_1,Y_1) = d_P(λ,μ).But in my problem, I want to fix the random variable X.$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