Displaying similar documents to “Integral of Complex-Valued Measurable Function”

The First Mean Value Theorem for Integrals

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

Similarity:

In this article, we prove the first mean value theorem for integrals [16]. The formalization of various theorems about the properties of the Lebesgue integral is also presented.MML identifier: MESFUNC7, version: 7.8.09 4.97.1001

The Measurability of Complex-Valued Functional Sequences

Keiko Narita, Noboru Endou, Yasunari Shidama (2009)

Formalized Mathematics

Similarity:

In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable...

Integral of Real-Valued Measurable Function 1

Yasunari Shidama, Noboru Endou (2006)

Formalized Mathematics

Similarity:

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

Egoroff's Theorem

Noboru Endou, Yasunari Shidama, Keiko Narita (2008)

Formalized Mathematics

Similarity:

The goal of this article is to prove Egoroff's Theorem [13]. However, there are not enough theorems related to sequence of measurable functions in Mizar Mathematical Library. So we proved many theorems about them. At the end of this article, we showed Egoroff's theorem.MML identifier: MESFUNC8, version: 7.8.10 4.100.1011

Lebesgue's Convergence Theorem of Complex-Valued Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2009)

Formalized Mathematics

Similarity:

In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences...

Fatou's Lemma and the Lebesgue's Convergence Theorem

Noboru Endou, Keiko Narita, Yasunari Shidama (2008)

Formalized Mathematics

Similarity:

In this article we prove the Fatou's Lemma and Lebesgue's Convergence Theorem [10].MML identifier: MESFUN10, version: 7.9.01 4.101.1015

The Lebesgue Monotone Convergence Theorem

Noboru Endou, Keiko Narita, Yasunari Shidama (2008)

Formalized Mathematics

Similarity:

In this article we prove the Monotone Convergence Theorem [16].MML identifier: MESFUNC9, version: 7.8.10 4.100.1011