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The DCT justifies exchanging the definite integral over $[a,b]$ with the infinite summation, allowing term-by-term integration. Specifically, it establishes that $\int_a^b \sum f_n(x)\,dx = \sum \int_a^b f_n(x)\,dx$.

Conditions: The previously stated convergence, domination, and finite-integral conditions are assumed to hold.; The summation is over the sequence of functions $f_n$.; The integration is over the closed interval $[a,b]$.

The DCT justifies exchanging the definite integral over $[a,b]$ with the infinite summation, allowing term-by-term integration. Specifically, it establishes that $\int_a^b \sum f_n(x)\,dx = \sum \int_a^b f_n(x)\,dx$.

Conditions: The previously stated convergence, domination, and finite-integral conditions are assumed to hold.; The summation is over the sequence of functions $f_n$.; The integration is over the closed interval $[a,b]$.

The three hypotheses are: (1) a sequence of functions $f_n$ converges to a function $f$ on the interval $[a,b]$; (2) there exists a dominating function $g$ such that $|g(x)| \ge |f_n(x)|$ for all $n$ and all $x \in [a,b]$; and (3) the integral of the dominating function over the interval is finite, meaning $\int_a^b g(x)\,dx < \infty$.

Conditions: The theorem is presented in a calculus-oriented form rather than a full measure-theoretic proof.; The interval $[a,b]$ is a closed, bounded real interval.

The three hypotheses are: (1) a sequence of functions $f_n$ converges to a function $f$ on the interval $[a,b]$; (2) there exists a dominating function $g$ such that $|g(x)| \ge |f_n(x)|$ for all $n$ and all $x \in [a,b]$; and (3) the integral of the dominating function over the interval is finite, meaning $\int_a^b g(x)\,dx < \infty$.

Conditions: The theorem is presented in a calculus-oriented form rather than a full measure-theoretic proof.; The interval $[a,b]$ is a closed, bounded real interval.