It's usual to integrate functions like $\int_\mathbb{R} \chi_E(x)\ dm(x)$, where $E$ is a Borel set and $m$ is the Lebesgue measure on $\mathbb{R}$. In this case we have $\int_\mathbb{R} \chi_E(x)\ dm(x) = m(E)$.
But, for example, consider the function $g:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that $g(x,t) = \chi_{(-\infty,x]}(t)$. Now we define $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(t) = \int_\mathbb{R} g(x,t)\ dm(x) = \int_\mathbb{R} \chi_{(-\infty,x]}(t)\ dm(x)$. Now the variable of the characteristic function is not the variable of integration. How can we integrate this?
Hint: Your function $g(x,t)$ is $1$ precisely when $t\leq x$, otherwise $0$. Therefore, it is $1$ precisely when $x\geq t$, otherwise zero. Therefore, it coincides with the function $g(x,t) = \chi_{[t,\infty)}(x)$. Can you go on now?