# How do we introduce a signed finite measure on the space of curves confined into the box $[0,1]^{n}$?

Given $$\Omega_{n} = \{\alpha:[0,1]\rightarrow[0,1]^{n}\,|\,\alpha\,\,\text{is smooth}\}$$, consider the equivalence relation: \begin{align*} & \alpha_{1} \sim \alpha_{2} \Leftrightarrow \int_{0}^{1}\langle G(\alpha_{1}(t)),\alpha^{\prime}_{1}(t)\rangle\,\mathrm{d}t = \int_{0}^{1}\langle G(\alpha_{2}(t)),\alpha^{\prime}_{2}(t)\rangle\,\mathrm{d}t,\\ \end{align*}

Where $$\alpha_{1}(0) = \alpha_{2}(0)$$ and $$\alpha_{1}(1) = \alpha_{2}(1)$$. It is assumed that $$G:[0,1]^{n}\rightarrow[0,1]^{n}$$ is known. Such set can naturally be considered as a metric space (thus a topological space) in accordance to the norm:

\begin{align*} \lVert\alpha\rVert = \max_{0 \leq t \leq 1}\lVert\alpha(t)\rVert_{2} \end{align*}

Let us define $$\Omega := \Omega_{n}/G$$ as the quotient space according to the above-mentioned equivalence relation. Thus we can introduce a topology on $$\Omega$$. Precisely speaking, the quotient topology: \begin{align*} \tau_{\Omega} := \{O\subset\Omega\,|\,\pi^{-1}(O)\in\tau_{\Omega_{n}}\} \end{align*}

Where $$\pi$$ is the map which associates each $$\alpha\in\Omega_{n}$$ to $$[\alpha]\in\Omega$$: $$\pi(\alpha) = [\alpha]$$. Finally, given the topological space $$(\Omega,\tau_{\Omega})$$, we can construct its associated Borel $$\sigma$$-algebra $$\Sigma$$.

Here is my question: how do we introduce a signed finite measure on $$(\Omega,\Sigma)$$? Precisely, I would like to define a triple $$(\Omega,\Sigma,\mathbb{P})$$ such that $$\mathbb{P}([\alpha]) = -\mathbb{P}([\alpha^{-}]) \geq 0$$, where $$\alpha^{-}(t) = \alpha(1-t)$$.

Such problem makes part of my research project on negative probabilities. I apologize if the question does not fit into Math Overflow context. Any help is appreciated. Thanks in advance.