[Math] When can the order of limit and integral be exchanged


  1. I was wondering for a real-valued function with two real variables, if there are some theorems/conclusions that can be used to decide the exchangeability of the order of
    taking limit wrt one variable and taking integral (Riemann integral,
    or even more generally Lebesgue integral ) wrt another variable,
    like $$\lim_{y\rightarrow a} \int_A f(x,y) \, dx = \int_A
    \lim_{y\rightarrow a} f(x,y) \,dx \text{ ?}$$
  2. If $y$ approaches $a$ as a countable sequence $\{y_n, n\in
    \mathbb{N}\}$, is the order exchangeable when $f(x,y_n), n \in \mathbb{N}$ is uniformly convergent in some subset for $x$ and $y$?
  3. How shall one tell if the limit and integral can be exchanged in the following examples? If not, how would you compute the values of the integrals:

    • $$\lim_{y\rightarrow 3} \int_1^2 x^y \, dx$$
    • $$ \lim_{y\rightarrow \infty} \int_1^2 \frac{e^{-xy}}{x} \, dx$$

Thanks and regards!

Best Answer

The most useful results are the Lebesgue dominated convergence and monotone convergence theorems.

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