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Let $BL^1_{loc}$ be the space of locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy<\infty$. Is this space complete ?

What I tried: Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of $BL^1_{loc}$ such that $$\sup_{x\in \mathbb{R}}\int_x^{x+1}|f_p(y)-f_q(y)|dy\to0,$$ when $p,q\to\infty$. Then, we have $$\int_{-N}^{N}|f_p(y)-f_q(y)|dy\to0,$$ for any $N$, so the sequence of functions $f_n$ is Cauchy in $L^1([-N,N])$ for each $N$, so we have an $L^1-$limit function $f$ for each compact interval, but I don't know if this function satisfies $$\sup_{x\in \mathbb{R}}\int_x^{x+1}|f_n(y)-f(y)|dy\to 0.$$I see that I didn't use the fact that we have uniformity for $x\in \mathbb{R}$.

user50618
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1 Answers1

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Yes, it's complete. First, observe that your norm is comparable to the simpler norm $$ \|f\| = \sup_{n\in\mathbb{Z}} \int_n^{n+1}|f(y)|\,dy \tag{1} $$ Indeed, $\|f\|\le |f|\le 2\|f\|$ because every interval of length $1$ is contained in some interval of the form $[n,n+2]$.

The space with $(1)$ is just the direct sum $\bigoplus_\infty X_n$ of Banach spaces $X_n=L^1([n,n+1])$. The direct sum preserves completeness, which is standard and proved here.

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