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.