transformation of two measures on different space

Question

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.

Let $\Sigma_U$ be the smallest $\sigma-$ field on $U$, and $\Sigma_E$ be the smallest $\sigma-$ field on $E$.

Let $M_U$ be the set of all measures on $(U,\Sigma_U)$, and $M_E$ be the set of all measures on $(E,\Sigma_E)$

For a measure $m_U\in M_U$, could we transform it as another $m_E \in M_E$?

In particular, for a probability measure on $(U,\Delta (U))$, could we transform it as another probability measure on $(E,\Delta (E))$?

Intuitively, I think the answer is yes but I cannot prove it, please help me.

Thanks a lot in advance.

Answer 1

The smallest $\sigma$-field on $U$ or $E$ is $\{\emptyset,U\}$ or $\{\emptyset,E\}$, there's only one probability measure on such a trivial thing.