[Math] What Does it Mean for a Function to have Finite Support

Question

I am working on a problem that states:

Let $f$ be integrable over $\mathbb{R}$ and $\varepsilon > 0$. Show that there is a simple function $\eta$ on $\mathbb{R}$ which has finite support and $\int_{\mathbb{R}} \lvert f – \eta \rvert < \varepsilon$

What does it mean for a simple function $\eta$ to have finite support? I do not need help with the actual problem, just the meaning. Unfortunately, Royden's fourth edition of 'Real Analysis' does not describe this.

Answer 1

It should mean

the function vanishes outside a set of finite measure

but not that only finitely many elements in the domain produce a nonzero value for the function.

We are talking about a measure theory book, after all. The main gauge on the size of a measurable set is its measure.

Of course, if the measure being used is the counting measure then it would turn into "only finitely many values of the domain produce nonzero values.