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Given totwo probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such that $f^*\mu = \gamma$?

Given to probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such that $f^*\mu = \gamma$?

Given two probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such that $f^*\mu = \gamma$?

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user2173168
user2173168
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Image of probability measures under measurable mappings

Given to probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such that $f^*\mu = \gamma$?