We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^infty$ onto the constants. Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.
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机译：我们调查了随机变量的期望值和条件期望值的可能定义，这些变量具有局部字段中的值，例如$ p $ -adic数字。我们通过类似于观察的方式定义期望，即对于$ L ^ 2 $中的实值随机变量，期望值是常数上的正交投影。先前的工作表明$ L ^ infty $的本地字段版本是$ L ^ 2 $的适当对应物，因此将本地字段值随机变量的期望值定义为其``投影''在$ L ^ infty $到常量中。与实际情况不同，生成的投影通常不是单个常数，而是局部域上度量中的一个球。但是，该期望操作的许多属性和相应的条件期望反映了实际值情况下熟悉的那些属性;例如，在适当的意义上，有条件期望是$ L ^ infty $的收缩，并且Tower属性保持不变。我们还定义了mar的相应概念，表明that的几个标准示例（例如，合适的独立随机变量的总和或乘积或由马尔可夫链组成的``谐波''函数）具有局部场类似物，并获得了可选抽样和mar收敛定理。
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