I have the following question.
We have a random variable $X:\Omega \to\mathbf{X} \subseteq \mathbb{R}^d$ and a continuous function $f:\mathbf{X}\to\mathbb{R}$ such that $f(X)$ and $X$ are both integrable. Introduce n-component partition of $\mathbf{X} \subseteq \mathbb{R}^d$ by $\mathbf{X}^{(n)} = \{\mathbf{X}^{(n)}(k): k =1,\dots,n\}$ and assume the generated sigma-algebra $\sigma^{(n)} = \sigma(\{X\in\mathbf{X}^{(n)}(k)\}:k=1,\dots,n)$ satisfies $\sigma(X) = \sigma(\cup_{n\in\mathbf{N}} \sigma^{(n)})$.
Then from the martingale convergence theorem and the continuous mapping theorem, we know that both random variables $E[f(X):\sigma^{(n)}]$ and $f(E[X:\sigma^{(n)}])$ converge to $f(X)$ almost surely.
My question is whether there exists a sequence of random variables $(Y^{(n)})_{n\in\mathbb{N}}$ where $\sup_{n\in\mathbb{N}} Y^{(n)}$ is integrable and $\lim_{n\to\infty}Y^{(n)} = 0$ almost surely such that $$ |f(X)| + Y^{(n)} \geq |f(E[X:\sigma^{(n)}])| $$ and $$ |E[f(X):\sigma^{(n)}]| + Y^{(n)} \geq |f(E[X:\sigma^{(n)}])| $$ holds???
Thank you.