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  "body": "\n## Summary\n\nA Markov kernel from a space $X$ to a space $Y$ assigns to each point of $X$ a [[probability measure|059TJN0NZ1]] on $Y$.\nThe assignment must be measurable in the point.\nIt generalises the [[transition matrix|0F462493EA]] of a Markov chain to arbitrary state spaces.\nKernels describe random transitions, conditional distributions and randomised maps.\nThey can be composed, and together with a starting measure they build measures on product spaces.\n\n## Explanations\n\n### L1 human\n\nA Markov kernel is a rule that says: if you are at point $x$, here is the probability distribution of where you go next.\nFor a finite set of states it is just a table whose rows are probability vectors.\nFor example, a random walk on $\\Z$ that moves to $x + 1$ or $x - 1$ with probability $\\tfrac12$ each is given by the kernel $\\kappa(x, \\cdot) = \\tfrac12 \\delta_{x+1} + \\tfrac12 \\delta_{x-1}$.\n",
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