This paper contains a development of the Theory of Lebesgue and Bochner
spaces of summable functions. It represents a synthesis of the results due to
H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford,
P. Halmos, and other contributors to this theory.
The construction of the theory is based on the notion of a measure on a
prering of sets in any abstract space X. No topological structure of the space
X is required for the development of the theory.
Assume that in a Lorentzian frame is given a relativistically admissible
trajectory of a point mass. An event in such a frame can be described by four
coordinates, first three representing the position and the last one the time of
the event. Let G denote the set of all events that do not lie on the
trajectory.
H. Cartan in his book on differential calculus proved a theorem generalizing
a Cauchy's mean-value theorem to the case of functions taking values in a
Banach space.
Cartan used this theorem in a masterful way to develop the entire theory of
differential calculus and theory of differential equations in finite and
infinite dimensional Banach spaces.
One of the highlight of this note is that the author presents the
relativistic gravity field that Einstein was looking for. The field is a
byproduct of the matter in motion. This field can include both the discrete and
continuous components. In free space the waves produced in this field propagate
with velocity of light.
Another highlight is the proof of amended Feynman's formulas for
electromagnetic potentials. This makes the formulas mathematically complete and
precise.
In a fixed Lorentzian frame given is a trajectory $r_2(t,r_0)$ of moving
plasma such that each line $t\mapsto r_2(t,r_0)$ of flow has derivatives with
respect to $t$ of order 3. The parameter $r_0$ represents the position of the
plasma at some initial time $t_0$ and changes in a compact set $F.$ It is
assumed that the map $(t,r_0)\mapsto r_2(t,r_0)$ is continuous and for fixed
$t$ the map $r_0\mapsto r_2(t,r_0)$ is one-to-one on $F.$
In a fixed Lorentzian frame given is a trajectory $r_2(t,r_0)$ of moving
plasma such that each line $t\mapsto r_2(t,r_0)$ of flow has derivatives with
respect to $t$ of order 3. The parameter $r_0$ represents the position of the
plasma at some initial time $t_0$ and changes in a compact set $F.$ It is
assumed that the map $(t,r_0)\mapsto r_2(t,r_0)$ is continuous and for fixed
$t$ the map $r_0\mapsto r_2(t,r_0)$ is one-to-one on $F.$
In this paper the author shows that, after replacing the ordinary derivative
with respect to time by partial derivative, and treating all quantities
appearing in Feynman's Law as fields, one can prove that the corrected formula
generates electromagnetic field satisfying Maxwell equations. The main result
of the paper is contained in the following Bogdan-Feynman theorem:
Given $n$ charges interacting with each other according to Feynman's law. Let
$(r_j(t),v_j(t))$ denote the position and velocity of the charge $q_j.$ The
list $y(t)$ of all such vectors is called a trajectory. A Lipschitzian
trajectory $x(t), (t\le0),$ with continuous derivative, on which the velocities
do not exceed some limiting velocity $v<c,$ where $c$ denotes the speed of
light, is called an initial trajectory. A locally Lipschitzian trajectory
$y(t)$ is called relativistically admissible if the velocities on it stay below
the speed of light $c.$