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  "body": "\n## Summary\n\nThe Bochner integral extends the Lebesgue integral to functions with values in a Banach space.\nIt is defined as a limit of integrals of simple functions, approximated in the mean.\nA strongly measurable function is Bochner integrable exactly when the integral of its norm is finite.\nIt keeps the main theorems of Lebesgue integration, such as dominated convergence.\nIt is the standard integral for vector-valued functions in analysis and partial differential equations.\n\n## Explanations\n\n### L1 human\n\nFor a function with vector values that takes finitely many values, the integral is a weighted sum of vectors.\nA general function is approached by such simple functions, closer and closer on average.\nThe Bochner integral is the limit of their integrals.\nFor example, a continuous curve $t \\mapsto u(t)$ in a Banach space, defined on $[0, 1]$, has a Bochner integral $\\int_0^1 u(t) \\, dt$, the limit of its Riemann sums.\n",
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