Dominated Convergence theorem

1. How do you use dominated convergence theorem?
2. How do you prove dominated convergence theorem?
3. What is meant by convergence theorem?

How do you use dominated convergence theorem?

The dominated convergence theorem states that “g” is a Lebesgue integrable function that ∣f_n∣ ≤ g nearly everywhere on I and for all n ∈ N. If lim_n→∞ ∫_I f_n(x) dx = ∫_I f(x)dx., then f is Lebesgue integrable on I.

How do you prove dominated convergence theorem?

Proof. Since the sequence is uniformly bounded, there is a real number M such that |f_n(x)| ≤ M for all x ∈ S and for all n. Define g(x) = M for all x ∈ S. Then the sequence is dominated by g.

What is meant by convergence theorem?

In real analysis, the monotone convergence theorem states that if a sequence increases and is bounded above by a supremum, it will converge to the supremum; similarly, if a sequence decreases and is bounded below by an infimum, it will converge to the infimum.