Let $(X_n)_{n \in \mathbb{N}}$ be independent random variables over a probability space $(\Omega,\mathfrak{F}, \mathbb{P})$
Is it true that then $(e^{X_n})_{n\in\mathbb{N}}$ are also independent random variables?
If yes, how would one conclude that
a) $(e^{X_n})_{n\in\mathbb{N}}$ are mesurable and
b) $(e^{X_n})_{n\in\mathbb{N}}$ are independent