definition of \(\mathcal{L}^p\) over measure space
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean topological space.
- The reader knows a definition of complex Euclidean topological space.
- The reader knows a definition of Borel \(\sigma\)-algebra of topological space.
- The reader knows a definition of function Lebesgue integrable over measurable subset of measure space.
- The reader knows a definition of %field name% vectors space.
- The reader admits the proposition that for any measure space, the union of any sequence of any locally negligible subsets is locally negligible.
Target Context
- The reader will have a definition of \(\mathcal{L}^p\) over measure space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( (M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\( \mathbb{C}\): \(= \text{ the complex Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\( \mathbb{R}\): \(= \text{ the Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\( F\): \(\in \{\mathbb{C}, \mathbb{R}\}\)
\( p\): \(\in \mathbb{R}\) such that \(1 \le p \lt \infty\)
\(*\mathcal{L}^p (M, A, \mu, F)\): \(= \{f: M \to F \vert \vert f \in \{\text{ the measurable functions }\} \land \vert f \vert^p \in \{\text{ the Lebesgue integrable functions }\}\}\), with the seminorm specified below, \(\in \{\text{ the seminormed } F \text{ vectors spaces } \}\)
\(*\mathcal{L}^\infty (M, A, \mu, F)\): \(= \{f: M \to F \vert f \in \{\text{ the measurable functions }\} \land \exists l \in \mathbb{R} (\forall m \in M (\vert f (m) \vert \lt l))\}\), with the seminorm specified below, \(\in \{\text{ the seminormed } F \text{ vectors spaces } \}\)
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Conditions:
For \(\mathcal{L}^p\), \(\Vert f \Vert = (\int \vert f \vert^p d \mu)^{1 / p}\)
\(\land\)
For \(\mathcal{L}^\infty\), \(\Vert f \Vert = inf \{r \in \mathbb{R} \vert 0 \le r \land \{m \in M \vert r \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\}\)
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2: Note
This is different from \(L^p (M, A, \mu, F)\).
Let us see that \(\mathcal{L}^p (M, A, \mu, F)\) is indeed a \(F\) vectors space.
\(c_1 f_1 + c_2 f_2: M \to F\) is measurable, as is well known.
\(\vert c_1 f_1 (m) + c_2 f_2 (m) \vert^p \le (\vert c_1 f_1 (m) \vert + \vert c_2 f_2 (m) \vert)^p\), but when \(\vert c_1 f_1 (m) \vert \le \vert c_2 f_2 (m) \vert\), \((\vert c_1 f_1 (m) \vert + \vert c_2 f_2 (m) \vert)^p \le (2 \vert c_2 f_2 (m) \vert)^p \le (2 \vert c_1 f_1 (m) \vert)^p + (2 \vert c_2 f_2 (m) \vert)^p\) and when \(\vert c_2 f_2 (m) \vert \lt \vert c_1 f_1 (m) \vert\), \((\vert c_1 f_1 (m) \vert + \vert c_2 f_2 (m) \vert)^p \le (2 \vert c_1 f_1 (m) \vert)^p \le (2 \vert c_1 f_1 (m) \vert)^p + (2 \vert c_2 f_2 (m) \vert)^p\), so, anyway, \((\vert c_1 f_1 (m) \vert + \vert c_2 f_2 (m) \vert)^p \le (2 \vert c_1 f_1 (m) \vert)^p + (2 \vert c_2 f_2 (m) \vert)^p = 2^p \vert c_1 \vert^p \vert f_1 (m) \vert^p + 2^p \vert c_2 \vert^p \vert f_2 (m) \vert^p\), so, \(\vert c_1 f_1 (m) + c_2 f_2 (m) \vert^p\) is Lebesgue integrable.
So, 1) and 6) of the requirements for the definition of %field name% vectors space are satisfied.
Let us see that the other requirements are satisfied.
2) \(\forall f_1, f_2 \in \mathcal{L}^p (M, A, \mu, F) (f_1 + f_2 = f_2 + f_1)\) (commutativity of addition): \(f_1 + f_2 = f_2 + f_1\).
3) \(\forall f_1, f_2, f_3 \in \mathcal{L}^p (M, A, \mu, F) ((f_1 + f_2) + f_3 = f_1 + (f_2 + f_3))\) (associativity of additions): \((f_1 + f_2) + f_3 = f_1 + (f_2 + f_3)\).
4) \(\exists 0 \in \mathcal{L}^p (M, A, \mu, F) (\forall f \in \mathcal{L}^p (M, A, \mu, F) (f + 0 = f))\) (existence of 0 element): \(0 \in \mathcal{L}^p (M, A, \mu, F)\) will do.
5) \(\forall f \in \mathcal{L}^p (M, A, \mu, F) (\exists f' \in \mathcal{L}^p (M, A, \mu, F) (f' + f = 0))\) (existence of inverse element): \(f' := - f\) will do.
7) \(\forall f \in \mathcal{L}^p (M, A, \mu, F), \forall r_1, r_2 \in F ((r_1 + r_2) . f = r_1 . f + r_2 . f)\) (scalar multiplication distributability for scalars addition): \((r_1 + r_2) . f = r_1 . f + r_2 . f\).
8) \(\forall f_1, f_2 \in \mathcal{L}^p (M, A, \mu, F), \forall r \in F (r . (f_1 + f_2) = r . f_1 + r . f_2)\) (scalar multiplication distributability for vectors addition): \(r . (f_1 + f_2) = r . f_1 + r . f_2\).
9) \(\forall f \in \mathcal{L}^p (M, A, \mu, F), \forall r_1, r_2 \in F ((r_1 r_2) . f = r_1 . (r_2 . f))\) (associativity of scalar multiplications): \((r_1 r_2) . f = r_1 . (r_2 . f)\).
10) \(\forall f \in \mathcal{L}^p (M, A, \mu, F) (1 . f = f)\) (identity of 1 multiplication): \(1 . f = f\).
Let us see that \(\mathcal{L}^\infty (M, A, \mu, F)\) is indeed a \(F\) vectors space.
\(c_1 f_1 + c_2 f_2: M \to F\) is measurable, as is well known.
\(\vert c_1 f_1 (m) + c_2 f_2 (m) \vert \le \vert c_1 f_1 (m) \vert + \vert c_2 f_2 (m) \vert = \vert c_1 \vert \vert f_1 (m) \vert + \vert c_2 \vert \vert f_2 (m) \vert \lt \vert c_1 \vert l_1 + \vert c_2 \vert l_2\).
So, 1) and 6) of the requirements for the definition of %field name% vectors space are satisfied.
Let us see that the other requirements are satisfied.
2) \(\forall f_1, f_2 \in \mathcal{L}^\infty (M, A, \mu, F) (f_1 + f_2 = f_2 + f_1)\) (commutativity of addition): \(f_1 + f_2 = f_2 + f_1\).
3) \(\forall f_1, f_2, f_3 \in \mathcal{L}^\infty (M, A, \mu, F) ((f_1 + f_2) + f_3 = f_1 + (f_2 + f_3))\) (associativity of additions): \((f_1 + f_2) + f_3 = f_1 + (f_2 + f_3)\).
4) \(\exists 0 \in \mathcal{L}^\infty (M, A, \mu, F) (\forall f \in \mathcal{L}^\infty (M, A, \mu, F) (f + 0 = f))\) (existence of 0 element): \(0 \in \mathcal{L}^\infty (M, A, \mu, F)\) will do.
5) \(\forall f \in \mathcal{L}^\infty (M, A, \mu, F) (\exists f' \in \mathcal{L}^\infty (M, A, \mu, F) (f' + f = 0))\) (existence of inverse element): \(f' := - f\) will do.
7) \(\forall f \in \mathcal{L}^\infty (M, A, \mu, F), \forall r_1, r_2 \in F ((r_1 + r_2) . f = r_1 . f + r_2 . f)\) (scalar multiplication distributability for scalars addition): \((r_1 + r_2) . f = r_1 . f + r_2 . f\).
8) \(\forall f_1, f_2 \in \mathcal{L}^\infty (M, A, \mu, F), \forall r \in F (r . (f_1 + f_2) = r . f_1 + r . f_2)\) (scalar multiplication distributability for vectors addition): \(r . (f_1 + f_2) = r . f_1 + r . f_2\).
9) \(\forall f \in \mathcal{L}^\infty (M, A, \mu, F), \forall r_1, r_2 \in F ((r_1 r_2) . f = r_1 . (r_2 . f))\) (associativity of scalar multiplications): \((r_1 r_2) . f = r_1 . (r_2 . f)\).
10) \(\forall f \in \mathcal{L}^\infty (M, A, \mu, F) (1 . f = f)\) (identity of 1 multiplication): \(1 . f = f\).
Let us see that \(\Vert f \Vert = (\int \vert f \vert^p d \mu)^{1 / p}\) is indeed a seminorm.
Let \(f_1, f_2 \in \mathcal{L}^p (M, A, \mu, F)\) and \(r \in F\) be any.
1) (\(0 \le \Vert f_1 \Vert\)) \(\land\) (\((f_1 = 0) \implies (0 = \Vert f_1 \Vert)\)): \(0 \le (\int \vert f_1 \vert^p d \mu)^{1 / p} = \Vert f_1 \Vert\); if \(f_1 = 0\), \(\Vert f_1 \Vert = (\int \vert f_1 \vert^p d \mu)^{1 / p} = (\int \vert 0 \vert^p d \mu)^{1 / p} = 0\).
2) \(\Vert r f_1 \Vert = \vert r \vert \Vert f_1 \Vert\): \(\Vert r f_1 \Vert = (\int \vert r f_1 \vert^p d \mu)^{1 / p} = (\int (\vert r \vert \vert f_1 \vert)^p d \mu)^{1 / p} = (\int \vert r \vert^p \vert f_1 \vert^p d \mu)^{1 / p} = (\vert r \vert^p \int \vert f_1 \vert^p d \mu)^{1 / p} = \vert r \vert (\int \vert f_1 \vert^p d \mu)^{1 / p} = \vert r \vert \Vert f_1 \Vert\).
Let us see that 3) \(\Vert f_1 + f_2 \Vert \le \Vert f_1 \Vert + \Vert f_2 \Vert\).
Let \(q\) be the exponent conjugate of \(p\).
\((\vert f_1 + f_2 \vert^{p - 1})^q = \vert f_1 + f_2 \vert^{(p - 1) q}\), but as \((p - 1) q = (1 - 1 / p) p q = 1 / q p q = p\), \(= \vert f_1 + f_2 \vert^p\), and as \(\int \vert f_1 + f_2 \vert^p d \mu \lt \infty\), \(\int (\vert f_1 + f_2 \vert^{p - 1})^q d \mu \lt \infty\), \(\vert f_1 + f_2 \vert^{p - 1} \in \mathcal{L}^q (M, A, \mu, F)\).
\(\vert f_1 + f_2 \vert^p = \vert f_1 + f_2 \vert \vert f_1 + f_2 \vert^{p - 1} \le (\vert f_1 \vert + \vert f_2 \vert) \vert f_1 + f_2 \vert^{p - 1}\).
So, \(\int \vert f_1 + f_2 \vert^p d \mu \le \int (\vert f_1 \vert + \vert f_2 \vert) \vert f_1 + f_2 \vert^{p - 1} d \mu = \int \vert f_1 \vert \vert f_1 + f_2 \vert^{p - 1} d \mu + \int \vert f_2 \vert \vert f_1 + f_2 \vert^{p - 1} d \mu \le \Vert f_1 \Vert_p \Vert \vert f_1 + f_2 \vert^{p - 1} \Vert_q + \Vert f_2 \Vert_p \Vert \vert f_1 + f_2 \vert^{p - 1} \Vert_q\), by the proposition that for any real number equal to or larger than \(1\) or \(\infty\), \(p\), its exponent conjugate, \(q\), any measure space, any element of \(\mathcal{L}^p (M, A, \mu, F)\) over the measure space, and any element of \(\mathcal{L}^q (M, A, \mu, F)\) over the measure space, the Lebesgue integral of the absolute product of the \(2\) elements is equal to or smaller than the product of the seminorms of the elements (Hölder's inequality).
\(= (\Vert f_1 \Vert_p + \Vert f_2 \Vert_p) \Vert \vert f_1 + f_2 \vert^{p - 1} \Vert_q = (\Vert f_1 \Vert_p + \Vert f_2 \Vert_p) (\int (\vert f_1 + f_2 \vert^{p - 1})^q d \mu)^{1 / q} = (\Vert f_1 \Vert_p + \Vert f_2 \Vert_p) (\int \vert f_1 + f_2 \vert^{(p - 1) q} d \mu)^{1 / q}\), but as \((p - 1) q = (1 - 1 / p) p q = 1 / q p q = p\), \(= (\Vert f_1 \Vert_p + \Vert f_2 \Vert_p) (\int \vert f_1 + f_2 \vert^p d \mu)^{1 / q}\).
So, \(\int \vert f_1 + f_2 \vert^p d \mu \le (\Vert f_1 \Vert_p + \Vert f_2 \Vert_p) (\int \vert f_1 + f_2 \vert^p d \mu)^{1 / q}\), so, \((\int \vert f_1 + f_2 \vert^p d \mu)^{1 - 1 / q} \le \Vert f_1 \Vert_p + \Vert f_2 \Vert_p\), but as \(1 - 1 / q = 1 / p\), the left hand side is \((\int \vert f_1 + f_2 \vert^p d \mu)^{1 / p} = \Vert f_1 + f_2 \Vert_p\), so, \(\Vert f_1 + f_2 \Vert_p \le \Vert f_1 \Vert_p + \Vert f_2 \Vert_p\).
So, \(\Vert f \Vert = (\int \vert f \vert^p d \mu)^{1 / p}\) is a seminorm.
Let us see that \(\Vert f \Vert = inf \{r \in \mathbb{R} \vert 0 \le r \land \{m \in M \vert r \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\}\) is indeed a seminorm.
It is valid, because \(\{r \in \mathbb{R} \vert 0 \le r \land \{m \in M \vert r \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\}\) is not empty, because the upper bound, \(l\), is in it.
While for \(\mathcal{L}^\infty (M, A, \mu, F)\), \(\Vert f \Vert\) is defined to be the infimum, let us see that it is in fact the minimum, which means that \(\{m \in M \vert \Vert f \Vert \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\).
Let \(s: \mathbb{N} \to \mathbb{R}\) be any sequence such that \(s (j) \in \{r \in \mathbb{R} \vert 0 \le r \land \{m \in M \vert r \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\}\) and \(s (j) - \Vert f \Vert \lt 1 / (j + 1)\), which is possible, because \(\Vert f \Vert\) is the infimum.
\(\{m \in M \vert \Vert f \Vert \lt \vert f (m) \vert\} = \cup_{j \in \mathbb{N}} \{m \in M \vert s (j) \lt \vert f (m) \vert\}\), because for each \(m \in \{m \in M \vert \Vert f \Vert \lt \vert f (m) \vert\}\), \(\Vert f \Vert \lt \vert f (m) \vert\), which means that \(\vert f (m) \vert = \Vert f \Vert + \delta\) for a \(\delta\) such that \(0 \lt \delta\), and there is a \(j \in \mathbb{N}\) such that \(s (j) - \Vert f \Vert \lt 1 / (j + 1) \lt \delta = \vert f (m) \vert - \Vert f \Vert\), which means that \(s (j) \lt \vert f (m) \vert\), so, \(m \in \{m \in M \vert s (j) \lt \vert f (m) \vert\}\); on the other hand, for each \(m \in \cup_{j \in \mathbb{N}} \{m \in M \vert s (j) \lt \vert f (m) \vert\}\), \(m \in \{m \in M \vert s (j) \lt \vert f (m) \vert\}\) for a \(j \in \mathbb{N}\), and \(\Vert f \Vert \le s (j) \lt \vert f (m)\), so, \(\Vert f \Vert \lt \vert f (m)\), so, \(m \in \{m \in M \vert \Vert f \Vert \lt \vert f (m) \vert\}\).
So, \(\{m \in M \vert \Vert f \Vert \lt \vert f (m) \vert\}\) is locally negligible, by the proposition that for any measure space, the union of any sequence of any locally negligible subsets is locally negligible.
Let \(f_1, f_2 \in \mathcal{L}^p (M, A, \mu, F)\) and \(s \in F\) be any.
1) (\(0 \le \Vert f_1 \Vert\)) \(\land\) (\((f_1 = 0) \implies (0 = \Vert f_1 \Vert)\)): the infimum is equal to or larger than \(0\), because \(0 \le r\); if \(f_1 = 0\), \(\{m \in M \vert r \lt \vert f (m) \vert\} = \{m \in M \vert r \lt \vert 0 \vert\}\) is empty for any non-negtive \(r\), so, the infimum is \(0\).
2) \(\Vert s f_1 \Vert = \vert s \vert \Vert f_1 \Vert\): when \(s = 0\), \(\Vert s f_1 \Vert = \Vert 0 \Vert = 0 = \vert s \vert \Vert f_1 \Vert\); otherwise, \(\Vert s f_1 \Vert = inf \{r \in \mathbb{R} \vert 0 \le r \land \{m \in M \vert r \lt \vert s f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\} = inf \{r \in \mathbb{R} \vert 0 \le r / \vert s \vert \land \{m \in M \vert r / \vert s \vert \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\}\), but as \(\Vert f_1 \Vert = inf \{r / \vert s \vert \in \mathbb{R} \vert 0 \le r / \vert s \vert \land \{m \in M \vert r / \vert s \vert \lt \vert f (m) \vert\} \in \{\text{ the locally negligible subsets of } M\}\}\), \(\Vert s f_1 \Vert = \vert s \vert \Vert f_1 \Vert\).
Let us see that 3) \(\Vert f_1 + f_2 \Vert \le \Vert f_1 \Vert + \Vert f_2 \Vert\).
Let \(S_j := \{m \in M \vert \Vert f_j \Vert \lt \vert f_j (m) \vert\}\), which is locally negligible.
Then, for each \(m \in M \setminus S_j\), \(\vert f_j (m) \vert \le \Vert f_j \Vert\), so, for each \(m \in (M \setminus S_1) \cap (M \setminus S_2)\), \(\vert f_j (m) \vert \le \Vert f_j \Vert\), but \((M \setminus S_1) \cap (M \setminus S_2) = M \setminus (S_1 \cup S_2)\), by the proposition that for any set, any subset minus the union of any subsets is the intersection of the 1st subset minus the 2nd chunk of subsets.
So, for each \(m \in M \setminus (S_1 \cup S_2)\), \(\vert f_1 (m) + f_2 (m) \vert \le \vert f_1 (m) \vert + \vert f_2 (m) \vert \le \Vert f_1 \Vert + \Vert f_2 \Vert\).
So, \(\{m \in M \vert \Vert f_1 \Vert + \Vert f_2 \Vert \lt \vert f_1 (m) + f_2 (m) \vert\} \subseteq S_1 \cap S_2\), but \(S_1 \cup S_2\) is locally negligible, by the proposition that for any measure space, the union of any sequence of any locally negligible subsets is locally negligible, so, \(\Vert f_1 + f_2 \Vert \le \Vert f_1 \Vert + \Vert f_2 \Vert\).
