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Given that $f_j, g_j$ and $f,g$ are locally integrable functions, i.e. they are in $L_{loc}^1(\mathbb{R})$.

Under the assumption that $f'_j(x)=g_j(x)$ in the sense of distributions, I wanna show that $f'(x)=g(x)$ in $\mathbb{R}$ in the sense of distributions.

Any hints?

Thanks in advance.

Jonathan Kiersch
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  • What's your definition of convergence in the space of locally integrable functions? Also I assume $f_j\to f,g_j\to g$ in this sense of convergence?! – Jonas Lenz Jul 27 '19 at 08:08
  • Yes. I define a function as locally integrable if is integral on any compact subset of R is finite – Jonathan Kiersch Jul 27 '19 at 10:06

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