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Can we apply the Dominated convergence theorem of Lebesgue to the sequence of functions $ f_n(x) =\frac{n}{x^2 +n^2}, \quad x\in \mathbb{R}$ ?

It is $ f_n(x) =\frac{n}{x^2 +n^2} \leq \frac{n}{2nx} =\frac{1}{2x}$ but the function $1/x$ is not Lebesgue integrable. How can I find a function $ g \in L^+( \mathbb{R})$ such that $|f_n| \leq g, n\in \mathbb{N}$ almost everywhere in $\mathbb{R}$ so I can use Lebesgue's theorem?

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    Is the conclusion of the Dominated Convergence Theorem true in this case? What is $\int_{-\infty}^\infty f_n(x)\ dx$?2012-02-02

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No, dominated convergence cannot be applied here. Indeed, we have

$\int_{-\infty}^\infty \frac{n\, dx}{x^2 + n^2} = \int_{-\infty}^\infty \frac{ d(x/n)}{(x/n)^2 + 1} = \int_{-\infty}^\infty \frac{dx}{x^2 + 1} = \pi$

eventhough pointwise the sequence converges to $0$.

Note also that $f_y(x) = \frac{y}{x^2+y^2}$ is the Poisson kernel for the upper half-plane. As such, it "converges to the dirac delta distribution" as $y \to 0$.

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    I knew the way define $g(x)=\sup_n |f_n(x)|$. Anyway thank's again!2012-02-02