Sigma Finite Measure restricted to a small sigma-algebra is still sigma finite? Let $(X,M,\upsilon)$ be a $\sigma$-finite measure, $N$ a sub-$\sigma$-algebra of $M$, then $\upsilon|_N$ is $\sigma$-finite measure in $(X,N)$?

No, consider $X=\mathbb{N}$ and $M$ is the collection of all the subsets of $\mathbb{N}$ and $v$ is usual counting measure. Now let $N=\\{\emptyset,\mathbb{N}\\}$ and you see the statement is false.