English
As in 147349, for integrable f, the equality holds between μ.withDensityᵥ f and the difference of signed measures built from positive/negative parts using ENNReal.ofReal.
Русский
Как и в 147349, для интегрируемой f равенство между μ.withDensityᵥ f и разностью знаковых мер, построенных по положительным/отрицательным частям через ENNReal.ofReal.
LaTeX
$$$\mu.withDensityᵥ f = \mathrm{toSignedMeasure}(\mu.withDensity (x \mapsto ENNReal.ofReal(f(x)))) - \mathrm{toSignedMeasure}(\mu.withDensity (x \mapsto ENNReal.ofReal(-f(x))))$$$
Lean4
/-- An indexing type to name each of the field axioms. The theory
of fields is defined as the range of a function `FieldAxiom ->
Language.ring.Sentence` -/
inductive FieldAxiom : Type
| addAssoc : FieldAxiom
| zeroAdd : FieldAxiom
| negAddCancel : FieldAxiom
| mulAssoc : FieldAxiom
| mulComm : FieldAxiom
| oneMul : FieldAxiom
| existsInv : FieldAxiom
| leftDistrib : FieldAxiom
| existsPairNE : FieldAxiom
