We can define your "anti-delta function" as an unusual type of distribution.

First, describe $F$ in terms of the integral $\langle F,\varphi\rangle:=\int_{-\infty}^{\infty}F(x)\,\varphi(x)\,dx$ for some test function $\varphi$. If I understand you correctly, you want this to be $\langle F,\varphi\rangle=\lim\limits_{a\to\infty}\frac{1}{2a}\int_{-a}^{a}\varphi(x)\,dx$, which is a linear functional of $\varphi$.

Distributions are defined as continuous/bounded linear functionals on a given space of test functions. There are various different spaces of distributions, depending on the domain of $\varphi$ used in their definition. The "anti-delta function" is not a distribution in the normal sense, because on the usual domains like $C_{c}^\infty$ or $C^\infty$ our functional $\langle F,\,\cdot\,\rangle$ is either identically zero, ill-defined or unbounded. To make $F$ a distribution, we need to choose another space of test functions $\varphi$, such that the limit $\lim\limits_{a\to\infty}\tfrac{1}{2a}\int_{-a}^{a}\varphi(x)\,dx$ is well-defined and continuous in $\varphi$.
  Possible test functions would include continuous functions on the one-point compactification $\mathbb{R}\cup\{\infty\}$, where $\langle F,\varphi\rangle=\varphi(\infty)$ (making it a "delta function at infinity")

We can define your "anti-delta function" as an unusual type of distribution.

First, describe $F$ in terms of the integral $\langle F,\varphi\rangle:=\int_{-\infty}^{\infty}F(x)\,\varphi(x)\,dx$ for some test function $\varphi$. If I understand you correctly, you want this to be $\langle F,\varphi\rangle=\lim\limits_{a\to\infty}\frac{1}{2a}\int_{-a}^{a}\varphi(x)\,dx$, which is a linear functional of $\varphi$.

Distributions are defined as continuous/bounded linear functionals on a given space of test functions. There are various different spaces of distributions, depending on the domain of $\varphi$ used in their definition. The "anti-delta function" is not a distribution in the normal sense, because on the usual domains like $C_{c}^\infty$ or $C^\infty$ our functional $\langle F,\,\cdot\,\rangle$ is either identically zero, ill-defined or unbounded. To make $F$ a distribution, we need to choose another space of test functions $\varphi$, such that the limit $\lim\limits_{a\to\infty}\tfrac{1}{2a}\int_{-a}^{a}\varphi(x)\,dx$ is well-defined and continuous in $\varphi$.

A way to construct viable function spaces and to simplify $\langle F, \varphi\rangle$ is to extend the domain of $\varphi$ to a compactification of $\mathbb{R}$. Possible test functions would include continuous functions on the one-point compactification $\mathbb{R}\cup\{\infty\}$, where $\langle F,\varphi\rangle=\varphi(\infty)$ (making it a "delta function at infinity").

Some of the other answers can be seen as variants of this: Using CDFs is equivalent to considering test functions $\varphi:\mathbb{R}\cup\{-\infty,\infty\}\to \mathbb{R}$ on the two-point compactification, where $\langle F,\varphi\rangle = c\,\varphi(-\infty)+(1-c)\,\varphi(\infty)$. Hyperfunctions are generalized distributions with holomorphic test functions.

There are various spaces of distributions,

We can define your "anti-delta function" as an unusual type of distribution.

First, describe $F$ in terms of the integral $\langle F,\varphi\rangle:=\int_{-\infty}^{\infty}F(x)\,\varphi(x)\,dx$ for some test function $\varphi$. If I understand you correctly, you want this to be $\langle F,\varphi\rangle=\lim\limits_{a\to\infty}\frac{1}{2a}\int_{-a}^{a}\varphi(x)\,dx$, which is a linear functional of $\varphi$.

Distributions are defined as continuous/bounded linear functionals on a given space of test functions. There are various different spaces of distributions, depending on the domain of $\varphi$ used in their definition. The "anti-delta function" is not a distribution in the normal sense, because on the usual domains like $C_{c}^\infty$ or $C^\infty$ our functional $\langle F,\,\cdot\,\rangle$ is either identically zero, ill-defined or unbounded. To make $F$ a distribution, we need to choose another space of test functions $\varphi$, such that the limit $\lim\limits_{a\to\infty}\tfrac{1}{2a}\int_{-a}^{a}\varphi(x)\,dx$ is well-defined and continuous in $\varphi$.
Possible test functions would include continuous functions on the one-point compactification $\mathbb{R}\cup\{\infty\}$, where $\langle F,\varphi\rangle=\varphi(\infty)$ (making it a "delta function at infinity")