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Let $(E,d)$ be a complete and separable metric space. Let $\mu$ be a probability measure on $E$. Let $f \colon E \rightarrow \mathbb{R}$ be continuous with $f \geq 0$. I want to find continuous and bounded functions $f_n \colon E \rightarrow \mathbb{R}$ with $f_n \leq f$ such that $$ \int_E f \,\mathrm{d}\mu = \sup_{n \in \mathbb{N}} \int_E f_n \,\mathrm{d}\mu. $$ Is this possible?

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$f_n=min(f,n)$ It looks straight forward. It is along the lines of the elementary description of Lebesgue integral.

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