I came across the following problem while doing a piece of research on automata theory.

Suppose we have a probability space $(\Omega, \mathcal{F}, \mu)$, where $\Omega$ is a set, $\mathcal{F}$ is a $\sigma$-algebra, and $\mu$ is a probability measure. Suppose we have **non-negative bounded** function $f:\Omega\rightarrow\mathbb{R}$. Suppose $A_i$, $i=1, 2,\dots$, is a sequence of sets in $\mathcal{F}$ that satisfy $\int_{A_i}f\ d\mu \leq a \mu\left(A_i\right)$ for each $i$ and $\mu\left(\bigcup_{i=1}^{\infty}A_i\right)=1$. My question is:

Does $\int_{\Omega}f\ d\mu$ is upper bounded by something **related to $a$**?

In my case, I also have $\mu(A_i)=p$ for all $i$, but I don't think the condition is essential to the question. However, if it is indeed essential or could simplify the proof, please feel free to use it.

The integral is bounded because $f$ is bounded and $\mu$ is a probability measure, but I need the integral to be bounded by something related to $a$, like a scalar of $a$.

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