# $L^p$ 空间基本知识1-关于 Lebesgue 积分的一些结论

### Beppo Levi 单调收敛定理

$$\sup_{n}\int f_n <\infty,$$

### Legesgue 控制收敛定理

a) $f_n(x)\rightarrow f(x)$ a.e. 收敛于 $\Omega$ 中,

b) 存在函数 $g\in L^1$, 使得每个 $n$, $|f_n(x)|\leqslant g(x)$, a.e. 于 $\Omega$ 中.

# 学习 Galois 理论须知的群论概念

Abelian Group. A group in which multiplication is commutative.

Alternating Group $A_n$. The subgroup of $S_n$ consisting of all the even permutations. it has order $\frac{1}{2}n!$.

Associativity. For all $x,y,z$, one has $(xy)z=x(yz)$. it follows that one does not need parentheses for any product of three or more factors.

Automorphism. An isomorphism of a group with itself.

Commutativity. For all $x,y$, one has $xy=yx$.