Then if is a -finite measure then this is an isometric isomorphism. There is a generalization that says the following.
Te map is an isometric injection is semifinite, that is, every measurable subset of (with possibly infinite -measure) has a futher finite -measure subset. Here, if is not semifinite, then there is a measurable subset with whose measurable subsets are all -measure zero. Then for functions supported on , such as , we have
So to these functions are “invisible” to functions. Semifiniteness is necessary and sufficient to break that invisibility.
The map is an isometric isomorphism is localizable, that is, semifinite and has a “gluing” property: for any family of measurable functions . such that