For any continuous function $h(x)$, if $\int_{-\infty}^{+\infty}h(x)f(x)=\int_{-\infty}^{+\infty}h(x)g(x)$, we say $f(x)$ is "specially equivalent to" g(x).
This is a concept our teacher proposed in the Signals and Systems course. It is used to analyze some properties of Dirac Function $\delta (t)$. eg:
$$x(t)\delta (t)"="x(0)\delta(t)$$
Another example is about the convolution:
If $x_1(t)"="x_2(t)$, then $x_1(t)\ast h(t)=x_2(t)\ast h(t)$.
Then it could be used to study the properties of LTI systems.
So what is the math background of this concept of "="(or is it just used in this specific field)? Is it related to Lebesgue measure?
This is called equality in the sense of measures. One can identify a measure $\mu$ with a the linear form acting on continuous functions $\varphi$ by the formula $\mu(\varphi) = \int \varphi(x) \,\mu(\mathrm d x)$ (This is called Riesz representation theorem). The functions $\varphi$ are called test functions. They are needed because a measure $\mu$ such as the Dirac delta cannot be defined pointwise (i.e. there is not meaning to $\delta_0(x)$ for a particular $x$) but there is a meaning to $\delta_0(\varphi) = \int \varphi(x) \,\delta_0(\mathrm d x) = \varphi(0)$.
Then, equality of two measures $\mu$ and $\nu$ is equivalent to write that for any continuous $\varphi$, $\mu(\varphi) = \nu(\varphi)$ (what you write as $\mu "=" \nu$). An usual function $f$ can be seen as a measure by identifying it with the measure $f = f(x)\,\mathrm d x$, where $\mathrm d x$ is the Lebesgue measure, i.e. it acts as a linear form as $f(\varphi) = \int \varphi(x)\,f(x)\,\mathrm d x$. This is consistant because if $f$ and $g$ are two functions, then $f(x) = g(x)$ for (almost) every $x$ is equivalent to $f(\varphi) = g(\varphi)$ for every continuous $\varphi$.
Another related notion is equality in the sense of distributions. The theory of distributions is a more general theory of generalized functions, where test functions are in $C^\infty_c$ (i.e. are infinitely smooth and compactly supported), and distributions are linear forms on $C^\infty_c$. If I write $\langle S,\varphi\rangle$ the action of a distribution $S$ on a function $\varphi\in C^\infty_c$, then two distributions $S$ and $T$ are equal in the sense of distributions iff $$ \langle S,\varphi\rangle = \langle T,\varphi\rangle $$ for any $\varphi\in C^\infty_c$. Measures can be seen as distributions by defining $\langle \mu,\varphi\rangle = \int \varphi(x)\,\mu(\mathrm d x)$, and so functions can be seen as distributions acting as $\langle f,\varphi\rangle = \int \varphi(x)\,f(x)\,\mathrm d x$. More general distributions that are not measures exist, such as the derivative of the Dirac delta, defined by $\langle \delta_0',\varphi\rangle = -\varphi'(0)$.