Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\{ A_i \}_{i\in I}$ be an at most countable partition of $\Omega$ (i.e. $I = \{1,\ldots, n\}$ or $I = \mathbb N$). Also let $X : \Omega \to \mathbb R$ be a random variable, define the random variable (suppose $P(A_i) > 0$) $$ Y := \sum_{i \in I} \frac{E[X \cdot I_{A_i}]}{P(A_i)} \cdot I_{A_i} $$ which is a function constant on each $A_i$ (indeed it is the conditional expectation $E[X | \sigma(\{ A_i \} : i \in I)]$, but this doesn't matter with respect to my question). Then if $X$ is integrable (i.e. $E[|X|] < \infty$), is then $Y$ integrable too, i.e. does $E[Y]$ exists?
If $I = \{1,\ldots, n\}$ we would have $$ E[|Y|] \le \int_{\Omega} \sum_{i=1}^n \left| \frac{E[X\cdot I_{A_i}]}{P(A_i)} \cdot I_{A_i} \right| dP = \sum_{i=1}^n \int_{\Omega} \left| \frac{E[X\cdot I_{A_i}]}{P(A_i)} \right|I_{A_i} dP = \sum_{i=1}^n \left| \frac{E[X\cdot I_{A_i}]}{P(A_i)} \right| P(A_i) = \sum_{i=1}^n |E[X\cdot I_{A_i}]| \le \sum_{i=1}^n \int_{A_i} |X| dP = E[|X|] < \infty $$ but for $I = \mathbb N$ I am not sure if all these rearrangement are valid, they would if we could apply the dominated convergence theorem, but this would require for each $n$ $$ \left| \sum_{i=1}^n \frac{E[X \cdot I_{A_i}]}{P(A_i)} \cdot I_{A_i}(\omega) \right| \le g(\omega) $$ for some integrable, i.e. $\int_{\Omega} |g| ~ dP < \infty$, function $g : \Omega \to \mathbb R$. But everything I can get here is that if $\omega \in A_j$ (which must be the case for some $A_j$ as this is a partition), then $$ \left| \sum_{i=1}^n \frac{E[X \cdot I_{A_i}]}{P(A_i)} \cdot I_{A_i}(\omega) \right| \le \left| \frac{E[X \cdot I_{A_j}]}{P(A_j)} \right| \le \frac{E[|X|]}{P(A_j)} $$ which does not make it easier, as for example showing that $g(\omega) := \sum_{i\in I} \frac{E[|X|]}{P(A_i)} I_{A_i}(\omega)$ is integrable does not seem simpler (without being able to get the sum out of the integral).