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If a function $f$ is bounded such that $|f|<M$, then can we say: $$ \left|\int_a^ \infty f(x)\, \mathrm{d}x \right| \le \int_a^ \infty |f(x)| \,\mathrm{d}x \le \int_a^ \infty M\, \mathrm{d}x$$ Is that correct?

Thank you!

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It is correct.

Since $ -|f| \leq f \leq |f|$: $$ -\int_a^\infty |f(x)|\,dx\leq \int_a^\infty f(x) \,dx \leq \int_a^\infty |f(x)| \,dx \\ \therefore \left| \int_a^\infty f(x) \,dx \right| \leq \int_a^\infty|f(x)| \,dx $$ The right one easily follows from the precondition.

$$ |f(x)| < M \Rightarrow |f(x)| \leq M \\ \int_a^\infty |f(x)|\, dx \leq \int_a^\infty M\,dx $$

Combine these two to get your desired inequality.

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