Why is $L^\infty \subset L^1$?

Question

Consider the identity function $f(\mathbf{x}) = 1$ on $\mathbb{R}^n$. It's clearly bounded but its $L^1$ norm $$\|f(\mathbf{x})\|_{L^1}= \int_{\mathbb{R}^n} d\mathbf{x}= \infty$$

I'm surely wrong, but I don't see the mistake.

Answer 1

You omitted the corresponding set $X \subset \mathbb{R}^n$ . If $X$ is bounded then indeed $L^\infty (X) \subset L^1(X)$ since $$ \int_{X} |f| dx \leq \int_{X} ||f||_\infty dx = ||f||_\infty \mu (X)< \infty $$ if $ f\in L^\infty(X)$

In general it is not true as your example pointed out correctly