Lagmental Vicfred

Uniform Convergence Permits Termwise Integration by Vicfred

tags: mathematics, uniform convergence, compactness, approximation

On a finite interval, uniformly convergent integrable functions have convergent integrals. A small computation will anchor the general statement before the abstraction takes over.

Definitions first

A sequence \(f_n:X\to\mathbf R\) converges uniformly to \(f\) when \(\sup_{x\in X}|f_n(x)-f(x)|\to0\). The supremum norm captures one error bound valid everywhere.

$$ f_n\to f\ \text{uniformly on }[a,b] $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \int_a^bf_n(x)\,dx\to\int_a^bf(x)\,dx $$

A small case in full

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \left|\int_a^b(f_n-f)\,dx\right|\le(b-a)\|f_n-f\|_\infty\to0 $$

The reusable statement

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_n\to f\ \text{uniformly on }[a,b],\\[5pt] \mathsf{C}\;&:\quad \int_a^bf_n(x)\,dx\to\int_a^bf(x)\,dx. \end{aligned} $$

A nearby false statement

Pointwise convergence allows the bad point to move with \(n\). Continuity, integration, and differentiation survive limits under different uniform hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int_a^bf_n(x)\,dx\to\int_a^bf(x)\,dx \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Fri 05 June 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.