What does 'supported in $ [a, b] $' mean in this theorem?

Question

What does 'supported in $ [a, b] $' mean in this theorem?

Theorem

Let $X=[a, b] \subset \mathbb{R},$ and let $\mathbb{X}$ be the collection of Borel sets in $X .$ Let $f:[a, b] \rightarrow[0, \infty)$ be continuous and supported in $[a, b] .$ Then $$ \int f d \lambda=\int_{a}^{b} f(x) d x $$ where the LHS denotes the Lebesgue integral and the RHS denotes the Riemann integral.

Answer 1

The support of a function is, more or less, the set on which the function takes non-zero values. The precise definition may vary a little, depending on the author or context (e.g. if we are considering functions on unstructured sets, then the support is precisely the complement of the zero set, while if we are considering functions on topological spaces, then the support is the closure of the complement of the zero set), but in the context of integration, a reasonable inference is that $$ \operatorname{supp}(f) = \overline{ \{x : f(x) \ne 0\} },$$ where $\operatorname{supp}(f)$ denotes the support of $f$.

If $\operatorname{supp}(f) \subseteq E$, then we may reasonably say that the function $f$ is supported in $E$. This implicitly means that $f(x) = 0$ whenever $x \not\in E$. In the context of the identity given in the question, this is important, as (in the context of Lebesgue integration on the real line) the notation $$ \int f \,\mathrm{d}\lambda $$ typically means "the integral of $f$ over the entire real line". By specifying that $f$ is supported in $[a,b]$, the author is, presumably, attempting to avoid the problem of $f$ being non-zero outside of $[a,b]$. Given that $f$ is explicitly only defined on $[a,b]$, this seems redundant, though perhaps the meaning is, as Crostul suggests, that $f$ should be extended to $\mathbb{R}$ by zero.

In any event, the given theorem is awkwardly stated. If it were me, I might rewrite it as

Theorem: Let $f : \mathbb{R}\to\mathbb{R}$ be a continuous function supported in $[a,b]$. Then $$ \int_{\mathbb{R}} f \,\mathrm{d}\lambda = \int_{a}^{b} f(x) \,\mathrm{d}x, $$ where the integral on the left-hand side is the Lebesgue integral, and the integral on the right-hand side is the Riemann integral.

Alternatively, we might suppose that $f : [a,b] \to\mathbb{R}$, and write $$ \int_{[a,b]} f\,\mathrm{d}\lambda \qquad\text{or}\qquad \int_{\mathbb{R}} f\chi_{[a,b]} \,\mathrm{d}\lambda, $$ where $\chi_{E}$ denotes the characteristic function of $E$.