KindiakMath

The Integrable Limit Theorem

One of the most fascinating explorations of integrable functions pertains its limiting conditions. When can limits of sequences of functions be integrable?

A sufficient condition is that if f_n \to f uniformly, then f must be integrable.

Theorem 1. Let \{f_n\} be a sequence of integrable functions f_n : [a, b] \to \mathbb R that converge point-wise to some function f : [a, b] \to \mathbb R. If f_n \to f uniformly, then f is integrable, and

\displaystyle \int_a^b f \equiv \int_a^b \lim_{n \to \infty} f_n = \lim_{n \to \infty} \int_a^b f_n.

Proof. Fix \epsilon > 0. Since f_n \to f uniformly, for any k_1 > 0, there exists N \in \mathbb N such that

j > N\quad \Rightarrow \quad \|f_j - f\|_\infty < k_1 \cdot \epsilon.

In particular, the estimate works for j = N+1. Since f_j is integrable, for any k_2 > 0, there exists a partition P \subseteq [a, b] such that

\displaystyle \sum_{i=1}^n (M_i(f_j, P) - m_i(f_j, P))\Delta x_i < k_2 \cdot \epsilon.

For any x \in [a, b], unraveling the estimate yields

\displaystyle f_j(x) - k_1 \cdot \epsilon< f(x) < f_j(x) + k_1 \cdot \epsilon,

which yields the estimates

\begin{aligned} M_i(f, P) &< M_i(f_j, P) + k_1 \cdot \epsilon,\\ m_i(f, P) &> m_i(f_j, P) - k_1 \cdot \epsilon. \end{aligned}

By unraveling the definitions and combining the estimates,

\begin{aligned} U(f, P) - L(f, P) &< U(f_j, P) - L(f_j, P) + 2k_1(b-a) \cdot \epsilon \\ &< k_2 \cdot \epsilon + 2k_1(b-a) \cdot \epsilon \\ &= (2k_1(b-a) + k_2) \cdot \epsilon.\end{aligned}

Setting k_1 = 1/(4(b-a)), k_2 = 1/2 yields the desired integrability result. Furthermore, by the triangle inequality for integrals, for any n > N,

\displaystyle \begin{aligned}\left| \int_a^b f_n - \int_a^b f \right| &= \left| \int_a^b (f_n - f )\right|\\ &\leq \int_a^b | f_n - f | \\ &\leq \int_a^b \| f_n - f \|_\infty \\ &\leq \| f_n - f \|_\infty \cdot (b-a) \\ &< k_1(b-a) \cdot \epsilon. \end{aligned}

Setting k_1 = 1/(b-a) yields the desired limit property

\displaystyle \int_a^b f \equiv \int_a^b \lim_{n \to \infty} f_n = \lim_{n \to \infty} \int_a^b f_n.

This is a classic example of valid instances when we can push the limit into the integral and vice versa. However, uniform convergence is typically too strong of a condition to be met, apart from some toy examples in standard tutorial problems.

With tools in measure theory, which we will explore a bit of next time, we can obtain a much more lenient condition. We will state this result, but not prove it, since its real utility shines forth in the context of measure theory.

Dominated Convergence Theorem. Let \{f_n\} be a sequence of Lebesgue-integrable functions f_n : [a, b] \to \mathbb R that converge point-wise to some function f : [a, b] \to \mathbb R.

If there exists a nonnegative Lebesgue-integrable function g : [a, b] \to \mathbb R such that |f_n| \leq g and |f| \leq g, then f is Lebesgue-integrable, and

\displaystyle \int_a^b f \equiv \int_a^b \lim_{n \to \infty} f_n = \lim_{n \to \infty} \int_a^b f_n.

Proof. Omitted. For further discussion in measure theory.

Clearly, we cannot prove what we don’t even know—what even is a Lebesgue-integrable function? The reason this condition is more relaxed than Theorem 1 is because bounded functions on [a, b] that are Riemann-integrable are Lebesgue-integrable too. Then the limit function f will be Lebesgue integrable.

From an applied perspective, probability theorists model random phenomena using random variables, which are defined to be Lebesgue-integrable (more technically, Lebesgue-measurable) functions. From probability theory arises all sorts of applications across various STEM fields.

While we won’t explore measure theory in its full glory in this discussion, we will briefly use our current discussions on Riemann-integrability to motivate the core objects of measure theory.

—Joel Kindiak, 24 Jan 25, 1452H

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