When can the order of limit and integral be exchanged?
- 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 $$\lim_{y\rightarrow a} \int_A f(x,y) \, dx = \int_A \lim_{y\rightarrow a} f(x,y) \,dx \text{ ?}$$
- 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 $y$?
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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!
Solution 1:
The most useful results are the Lebesgue dominated convergence and monotone convergence theorems.
Solution 2:
@Tim: You wrote in a comment: "For both theorems you mentioned, they apply to a discrete sequence of functions. In my questions, the index is continuous. How would that be coped with?"
If $\lim_{y\to a} f(x,y)$ exists, then $\lim_{n\to\infty} f(x,y_n)$ exists, for every sequence $\{y_n\}_{n=1}^\infty$ that approaches $y$, and conversely. You can use that to show that the dominated convergence theorem and the monotone convergence theorem still work in the "continuous" setting.