In search of a literature or theory that deals with $\int_{[a,b]}f(x)dx \geq \int_{g([a,b])}f(x)dx$ I stumble upon the following problem. Given an increasing function $g$ is there a function $f$ (not identically null) s.t.
$$\int_{[a,b]}f(x)dx \geq \int_{g([a,b])}f(x)dx$$
And, if there is, what can we say about $f$. The problem could be reinstated in terms of pushforward, i.e. find a signed measure $\mu$ s.t. $\mu([a,b]) \geq g_{*}(\mu)([a,b])$. There are also other analogous ways to approach it (in terms of differential equation the problem is a composition of function). However, none of this seems to shed light upon the problem. I tried to find a theory of some sort that could help me with it, but so far have not been successful.
$\textbf{Edit:}$ Considering avs answer the following idea emerged. Let $g$ be a strictly increasing continuouosly differentiable function. Let $A = \{x : g'(x) < 1\}$ and $B = \{x : g'(x) > 1\}$. Then $g$ is a contraction in every open interval in $A$ (actually, a subcontraction) and is an expansion (in the sense that its inverse is a contraction) in every open interval in $B$. Thus let $f$ be s.t. $f \leq 0$ in A, $f \geq 0$ in $B$ and $f = 0$ in $(A \cup B)^c$. Then $f$ must be a solution.
 A: A simple experiment gives a less than encouraging result: let
$$
g(x) = 2x, \quad a = 0, \quad b > 0.
$$
It follows that
$$
g([a, b]) = g([0, b]) = [0, 2b].
$$
Now imagine a measure $\mu$ on the (Borel $\sigma$-algebra of) $\mathbb{R}$ satisfies the required inequality.  Then we would need to have
$$
\mu([0, b]) \geq \mu ( g ([0, b]) ) = \mu([0, 2b]) = \mu([0, b]) + \mu(]b, 2b]),
$$
which would imply that
$$
\mu\left(]b, 2b]\right) = 0 \mbox{ for all positive }b.
$$
Thus, the existence of a measure $\mu$ with the desired property (a, sort of, "monotonicity w.r.t. $g$") depends a lot on what $g$ is.
If $g$ is a simple contraction, e.g., $g(x) = x / 2$, then it is not hard to show that $\mu$ can be taken to be the Lebesgue measure on $\mathbb{R}$.
It's also not that hard to construct a $\mu$ that will fail the desired property for the latter $g$; e.g., let $\mu$ have the density function
$$
f(x) = {C \over 1 + x^2}, \quad \mbox{where $C$ is large enough}.
$$
Basically, $f(x)$ assigns large measures to subsets near $0$, and small ones far away from $0$, but $g$ takes subsets that are far from $0$ and brings them closer to $0$, thus increasing their measure.
If you are looking for the field of mathematics that studies this, I'd try to look at measures as linear functionals.  The general concept of monotonicity of functionals (which are generally not in a Euclidean field, hence devoid of a "natural" linear ordering) is usually developed by using the concept of a cone.
