Levi-Civita域上的可测函数空间
Spaces of measurable functions on the the Levi-Civita field
摘要：
本文在Levi-Civita域$\mathcal{R}$上引入了其$p$次幂关于一致测度可和的可测函数的$\mathcal{L}^p$空间.这些空间是基于勒贝格测度的实$L^p$空间的对应物。然而，它们缺乏$L^p$空间的一些性质：例如，$\mathcal{L}^p$空间关于$p$ -范数是不完备的。这激发了对$\mathcal{L}^p$空间关于强收敛的完备化的研究，记为$\mathcal{L}_s^p$。事实证明，$\mathcal{L}_s^p$空间是Banach空间，并且可以在$\mathcal{L}_s^2$上定义内积，从而使其成为希尔伯特空间。尽管有这些积极的结果，这些空间仍然不够丰富，以表示每一个真正的连续函数。为此，我们解决了在$\mathcal{R}$的弱收敛的措施的可测函数序列的实可测函数的表示。我们还定义了一个对偶之间的可测函数和连续函数的代表。这种对偶性使得研究某些代表实分布的可测函数成为可能。我们集中讨论Dirac分布的代表和著名的Dirac分布与Heaviside分布之间的乘积问题，并证明了用$\mathcal{R}$上的可测函数得到的解与用其它非线性广义函数得到的结果是一致的.
英文摘要：
We introduce the $\mathcal{L}^p$ spaces of measurable functions whose $p$-th power is summable with respect to the uniform measure over the Levi-Civita field $\mathcal{R}$. These spaces are the counterparts of the real $L^p$ spaces based upon the Lebesgue measure. Nevertheless, they lack some properties of the $L^p$ spaces: for instance, the $\mathcal{L}^p$ spaces are not complete with respect to the $p$-norm. This motivates the study of the completions of the $\mathcal{L}^p$ spaces with respect to strong convergence, denoted by $\mathcal{L}_s^p$. It turns out that the $\mathcal{L}_s^p$ spaces are Banach spaces and that it is possible to define an inner product over $\mathcal{L}_s^2$, thus making it a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of real measurable functions as sequences of measurable functions in $\mathcal{R}$ that weakly converge in measure. We also define a duality between measurable functions and representatives of continuous functions. This duality enables the study of some measurable functions that represent real distributions. We focus our discussion on the representatives of the Dirac distribution and on the well-known problem of the product between the Dirac and the Heaviside distribution, and we show that the solution obtained with measurable functions over $\mathcal{R}$ is consistent with the result obtained with other nonlinear generalized functions.
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