Let $T$ be a non-negative random variable.
Why is it true that
$$\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$$
According to me it would make sense that $\mathbb{E}[1(T=k)]=\mathbb{P}[T=k]$.
So where is my mistake or is the statement above false?
More Context:
When $X_i$ are i.i.d. random variables, independent of $T$ with $\mathbb{E}[T_i]=\mu$, $Y_i:=|X_i|$, then we have for $S_k:=\sum_{i=1}^k X_i$ that
$$\mathbb{E}[|S_{T}|]\leq \sum_{k=1}^{\infty} \mathbb{E}[Y_k]\mathbb{E}[1(T=k)]=\mathbb{E}[Y_1]\sum_{k=0}^{\infty} k \mathbb{P}[T=k]=\mathbb{E}[Y_1]\mathbb{E}[T].$$
Maybe I misinterpret it and the question I ask about is wrong, or the master solution is wrong.