I consider a special kind of sets in $\mathbb{R}^n_+$ given by $G_t = $ {$x \in \mathbb{R}^n_+ \mid g(x) < t$}, where $\nabla g > 0$ entrywise. Let's consider an integral $$ f(t) = \int\limits_{ G_t } \mu(dx) $$ If measure $\mu$ is absolutely continious with the density $a(x)$ and $\Omega$ is a $(n-1)$-form such that $dg \wedge \Omega = dx$ than $$ f(t) = \int\limits_{ G_t } a(x) dx = \int\limits_{ G_t } a(x) dg \wedge \Omega $$ and $$ \frac{\Delta f}{\Delta t} = \frac{1}{\Delta t} \int\limits_{ t \leq g(x) < t + \Delta t } a(x) dg \wedge \Omega \to \int\limits_{ g(x) = t } a(x) \Omega = \frac{df}{dt} $$ Help me please with the generalisation of this result to the case of arbitrary measures. I think that flows currents (continious linear functionals on the set of differential forms) must be involved (I try to find a similar representation for the derivative $f'(t)$).
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Derivative of the Lebesgue integral. FlowsCurrents. |
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Derivative of the Lebesgue integral. Flows.I consider a special kind of sets in $\mathbb{R}^n_+$ given by $G_t = $ {$x \in \mathbb{R}^n_+ \mid g(x) < t$}, where $\nabla g > 0$ entrywise. Let's consider an integral $$ f(t) = \int\limits_{ G_t } \mu(dx) $$ If measure $\mu$ is absolutely continious with the density $a(x)$ and $\Omega$ is a $(n-1)$-form such that $dg \wedge \Omega = dx$ than $$ f(t) = \int\limits_{ G_t } a(x) dx = \int\limits_{ G_t } a(x) dg \wedge \Omega $$ and $$ \frac{\Delta f}{\Delta t} = \frac{1}{\Delta t} \int\limits_{ t \leq g(x) < t + \Delta t } a(x) dg \wedge \Omega \to \int\limits_{ g(x) = t } a(x) \Omega = \frac{df}{dt} $$ Help me please with the generalisation of this result to the case of arbitrary measures. I think that flows (continious linear functionals on the set of differential forms) must be involved (I try to find a similar representation for the derivative $f'(t)$).
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