Consider a sequence of Lebesgue measurable sets $(M_i)$, such that $M_i\cap M_j=\emptyset \forall i\neq j$. Prove that $\sum_i a_i\chi_{M_i}$ converges in the $L_p$ spaces if and only if $\sum_n |a_i|^pm(M_i)<\infty.$
Not sure where to start on this one. A hint to begin would be helpful. The second part of the statement kind of looks like the norm that is used in $L_p$ space (not exactly but somewhat similar if that is relevant?), i.e, $$||f||_p=\left(\int_E |f|^p\right)^{1/p}$$
But I don't see the relevance of the sets being disjoint. Can someone give me a prod in the right direction?
Here's a further hint based of David C Ullrich's comment, in case you need it:
Fix $n<m\in\mathbb{Z}^+$. Then:
$$\sum_{i=1}^n \alpha_i\chi_{M_i}-\sum_{i=1}^m\alpha_i\chi_{M_i}=\sum_{i=n+1}^m\alpha_i\chi_{M_i}$$
By disjointness, we see that this function takes the value $\alpha_i$ on $M_i$, and $0$ on the complement of $\bigcup_{i=n+1}^m M_i$.
Therefore:
$$\left(\sum_{i=n+1}^m\alpha_i\chi_{M_i}\right)^p=\sum_{i=n+1}^m\alpha_i^p\chi_{M_i}$$
and hence, by the same token:
$$\left\| \sum_{i=1}^n \alpha_i\chi_{M_i}-\sum_{i=1}^m\alpha_i\chi_{M_i}\right\|_p^p=\int \left|\sum_{i=n+1}^m\alpha_i\chi_{M_i}\right|^p \ dm=\int\sum_{i=n+1}^m |\alpha_i|^p\chi_{M_i}\ dm=\sum_{i=n+1}^m|\alpha_i|^pm(M_i)$$
EDIT: Regarding the usefulness of the disjointness of the $M_i$, call the entire space $X$. If $x\in X$, there's two possibilities: either $x\in \bigcup_{i=n}^mM_i$ or $x\not \in \bigcup_{i=n}^m M_i$. In the latter case, $\sum_{i=n}^m \alpha_i \chi_{M_i}(x)=0$ since each summand is $0$.
Now, if $x\in \bigcup_{i=n}^m M_i$, then there exists $j\in \{n,n+1,\dots,m\}$ such that $x\in M_j$. Furthermore, since the $\{M_i\}$ are disjoint, said number $j$ is unique.
We've thus proved:
$$ \sum_{i=n}^m\alpha_i \chi_{M_i}(x)=\begin{cases} \alpha_j & \text{if $x\in M_j$}\\ 0 & \text{if $x\not\in \bigcup_{i=n}^m M_i$} \end{cases} $$
Hence
$$ \left(\sum_{i=n}^m\alpha_i \chi_{M_i}(x)\right)^p=\begin{cases} \alpha_j^p & \text{if $x\in M_j$}\\ 0 & \text{if $x\not\in \bigcup_{i=n}^m M_i$} \end{cases} $$