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Anthony
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Setting

  • For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R}).$$

  • Assume that all the functions involved are striclty positive(1) striclty positive and that for any $i \in \mathbb{N}$, we have (2) $$ 0 < \int_\mathbb{R} f_i^2 g_i^{-1} < \infty$$ and also (3) $$ 0 < \int_\mathbb{R} f^2 g^{-1} < \infty $$

Question

Do we have the following convergence $$\int_\mathbb{R} f_i^2 g_i^{-1} \rightarrow \int_\mathbb{R} f^2 g^{-1} $$

Setting

  • For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R}).$$

  • Assume that all the functions involved are striclty positive and that for any $i \in \mathbb{N}$, we have $$ 0 < \int_\mathbb{R} f_i^2 g_i^{-1} < \infty$$ and also $$ 0 < \int_\mathbb{R} f^2 g^{-1} < \infty $$

Question

Do we have the following convergence $$\int_\mathbb{R} f_i^2 g_i^{-1} \rightarrow \int_\mathbb{R} f^2 g^{-1} $$

Setting

  • For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R}).$$

  • Assume that all the functions involved are (1) striclty positive and that for any $i \in \mathbb{N}$, we have (2) $$ 0 < \int_\mathbb{R} f_i^2 g_i^{-1} < \infty$$ and also (3) $$ 0 < \int_\mathbb{R} f^2 g^{-1} < \infty $$

Question

Do we have the following convergence $$\int_\mathbb{R} f_i^2 g_i^{-1} \rightarrow \int_\mathbb{R} f^2 g^{-1} $$

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Anthony
  • 125
  • 6

$L^1$ convergence

Setting

  • For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R}).$$

  • Assume that all the functions involved are striclty positive and that for any $i \in \mathbb{N}$, we have $$ 0 < \int_\mathbb{R} f_i^2 g_i^{-1} < \infty$$ and also $$ 0 < \int_\mathbb{R} f^2 g^{-1} < \infty $$

Question

Do we have the following convergence $$\int_\mathbb{R} f_i^2 g_i^{-1} \rightarrow \int_\mathbb{R} f^2 g^{-1} $$