In the proposition below from Measure Theory and Probability by Athreya and Lahiri, DCT was used to justify the existence of $t$ in the first line of the proof.
But I can't think how this was applied from the DCT as provided in page 7:
Can somebody explain how this is so?
$$\lim_{t\to\infty}|f(x)|\chi_{\{|f|>t\}}=0 ~~~~\text{a.e.}$$
(Because $f\in L^1$ if we had a set of positive measure that was infinite, the integral would be infinite as well)
And $|f(x)|\chi_{\{|f|>t\}}$ is dominated by an integrable function, namely $|f|$ (as $f\in L^1$), so we have:
$$\lim_{t\to\infty} \int_{|f|>t}|f|~d\mu=\lim_{t\to\infty} \int_{\Omega} |f(x)|\chi_{\{|f|>t\}} ~d\mu=\int_{\Omega}0~d\mu=0$$
Then just apply definition of convergence of numbers.