Show that $EX_1 1_A \geq 0$ given A is an event $\left(\sum\limits_1^n X_i>0 \right)$

Question

Let A be the event

$\left(\sum\limits_{i=1}^n X_i>0 \right)$ where $\{X_i\}_{i = 1, 2, ..., n}$ is an independent and identically distributed collection of random variables

Show that $E[X_1 \mathbb{1}_A] \geq 0$

Previous part:

the first two parts of the question which I solved were to show how two probabilities agree on a Pi-system agree on the sigma-algebra it generates and a sequence of i.i.d random variables's joint distribution are equal no matter what permutation it is in. Not sure if this helps

Answer 1

Since $X_1, \dots, X_n$'s are iid,

$$E[X_1 1_A] = E[X_k 1_A] \ \forall 1 \le k \le n$$

$$\to \sum_k E[X_k 1_A] = \sum_k E[X_1 1_A] = nE[X_1 1_A]$$

Also, $$\sum_k E[X_k 1_A] = E[(\sum_k X_k) 1_A] = \int_A \sum_k X_k dP$$

Now $$\int_A \sum_k X_k dP > 0$$

by applying monotonicity of integration on $$(\sum_k X_k)1_A > 0$$