Pushforward measure

Good evening,

Let $$\mu$$ and $$\nu$$ two measures on $$X$$ and $$Y$$.

Do you know when it exists a measurable function $$h : X \rightarrow Y$$ such as $$\nu = h\text{#}\mu$$ with $$h\text{#}\mu(B) = \mu(h^{-1}(B))$$.

Thank you, have a good day.

• Good evening. What the hashtag mean ? – user486983 May 25 at 5:03
• I tried to explain in my post, it is called pushfoward measure. – CechMS May 31 at 15:36
• maybe you can apply the theorem of Radon-Nikodym (Wikipedia) in some way; if you had a function $f : X \to Y$ such that $f \# \mu \ll \nu$, then this theorem tells you, there is such an $h$ you're looking for. – Targon Jun 6 at 9:03
• Hi ! I beg your pardon, I thought there was no answer (and there isn't still) so I didn't check but there was a comment. I'm not sur it works. Can you explain please ? – CechMS Jun 23 at 16:09

Here's two ideas to begin.

1) For exemple if $$X = Y = \{0,1\}$$, $$\mu = 1/3 \delta_{0} + 2/3 \delta_{1}$$, $$\nu = 1/2(\delta_{0} + \delta_{1})$$ then there is no such application.

2) If $$X = Y = R^{n}$$ and $$\mu := f.\mathcal{L}$$ et $$\mu$$ two probabilties then it exists $$\phi$$ convex such as $$\nabla{\phi} \text{#} \mu = \nu$$.

I think I have solution for discrete measures. Let $$\mu = \sum_{i=1}^{n} \delta_{x_{i}} a_{i}$$ and $$\nu = \sum_{i=1}^{n} \delta_{y_{i}} b_{i}$$.

A such $$h$$ exists if, and only if, $$b_{i} = \sum_{J(i)} a_{j}$$ with $$J(i)$$ a disjoint union of $$\{1,...,n\}$$. In that case $$h$$ is easy to compute.

Do you agree ?

I was told it's, in general, a really hard problem. So I would feel glad if someone has references (books, articles, ...) or maybe can show me some speciale cases :).

Thank you for your help, have a nice day.

On this thread : When does a measurable function exist with a given distribution?

@MichaelGreinecker gave us a proof of the following result : Let $$(A,\mathcal{A},P)$$ an atomless probability space, $$(B,\mathcal{B},\mu)$$ a polish space.

It exists a mesurable function $$h:A \rightarrow B$$ such as $$h\text{#}P= \mu$$