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Let

  1. $a_1 : [B_1,B_2]\to\mathbb{R}^+$ be continuous and increasing.
  2. $a_2 : (B_1,B_2]\to\mathbb{R}^+$ be continuous and decreasing with $\lim_{x\to B_1^+} a_2(x) = \infty$ and $a_1(x) \leq a_2(x)$.
  3. $f : \mathbb{R}^3\to\mathbb{R}^+$ be continuous in all its arguments and bounded.

When is $$ \lim_{x\to B_1^+} \int_{a_1(x)}^{a_2(x)}f(s,x,a_2(x))ds = \int_{a(B_1)}^\infty f(s, B_1, \infty)ds? $$

The dominated or monotone convergence theorems do not seem to help because the limits of the integral are also functions of the limiting sequence.

1 Answers 1

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Write $$ \int_{a_1(x)}^{a_2(x)}f(s,x,a_2(x))\,ds=\int_{a_1(B_1)}^{\infty}\chi_{[a_1(x),a_2(x)]}(s)\,f(s,x,a_2(x))\,ds. $$ To apply a convergence theorem you will need further assumptions on $f$, like:

  • $f(s,y,z)$ converges as $z\to\infty$.
  • $|f(s,x,a_2(x))|\le h(s,x)$ and $\int_{a_1(B_1)}^\infty h(s,x)\,dx<\infty$ for all $x$
  • A monotonicity condition
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    Thanks! the three assumptions you mentioned are satisfied in my problem. Could you please give me a reference to the theorem that can be applied? – 2017-01-20
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    You can use the dominated convergence theorem. By the way, $\chi_A$ is the characteristic function of $A$. – 2017-01-20
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    Please see my corrected question. I made a mistake in the variable of integration. The correct variable of integration is $s$. – 2017-01-20
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    I have edited my answer accordingly. – 2017-01-20