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Phil Isett
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The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure -- such functions are called ``bounded variation''. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass (the case $f'$ being in $L^1 $ is exactly when $f$ is absolutely continuous). One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.

The step to prove it for smooth compactly supported functions may be done by applying the dominated convergence theorem to

$\int (f(x+h) - f(x))/h~ dx$

as $h$ tends to $0$. I think that no matter how you try to prove the fundamental theorem, the mean value theorem will enter in somewhere (here it enters to bound the difference quotients).

The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass. One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.

The step to prove it for smooth compactly supported functions may be done by applying the dominated convergence theorem to

$\int (f(x+h) - f(x))/h~ dx$

as $h$ tends to $0$. I think that no matter how you try to prove the fundamental theorem, the mean value theorem will enter in somewhere (here it enters to bound the difference quotients).

The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure -- such functions are called ``bounded variation''. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass (the case $f'$ being in $L^1 $ is exactly when $f$ is absolutely continuous). One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.

The step to prove it for smooth compactly supported functions may be done by applying the dominated convergence theorem to

$\int (f(x+h) - f(x))/h~ dx$

as $h$ tends to $0$. I think that no matter how you try to prove the fundamental theorem, the mean value theorem will enter in somewhere (here it enters to bound the difference quotients).

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Phil Isett
  • 2.2k
  • 1
  • 24
  • 27

The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass. One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.

The step to prove it for smooth compactly supported functions may be done by applying the dominated convergence theorem to

$\int (f(x+h) - f(x))/h~ dx$

as $h$ tends to $0$. I think that no matter how you try to prove the fundamental theorem, the mean value theorem will enter in somewhere (here it enters to bound the difference quotients).

The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass. One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.

The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass. One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.

The step to prove it for smooth compactly supported functions may be done by applying the dominated convergence theorem to

$\int (f(x+h) - f(x))/h~ dx$

as $h$ tends to $0$. I think that no matter how you try to prove the fundamental theorem, the mean value theorem will enter in somewhere (here it enters to bound the difference quotients).

Source Link
Phil Isett
  • 2.2k
  • 1
  • 24
  • 27

The one-dimensional fundamental theorem (with a Lebesgue integral on the right hand side) holds in greatest generality when the derivative $f'$ (taken in the sense of distributions) is a measure. For example, if $f$ is nondecreasing, it is measurable, locally bounded and therefore a distribution. By taking difference quotients, $f'$ is a non-negative distribution and hence a measure (as one can show with the Riesz Representation theorem). You can prove the formula

$f(b) - f(a) = \int_a^b f'(x) dx$

at points $a$ and $b$ to which the measure $f'(x)$ does not assign positive mass. One proof is by convolving with a mollifier, quoting the result for smooth functions, then put the dual mollifier on the characteristic function of [a,b]. The mollifier converges to the characteristic function everywhere but the endpoints, and, say, the dominated convergence theorem allows you to take the limit as long as $a$ and $b$ do not have positive mass with respect to $f'$. But if you take the Heaviside function, its derivative is a delta function, and the formula essentially fails if you try to use $0$ as an endpoint. However, even in cases like this one the formula works for any $a, b$ not equal to $0$.