Probability – what does this notation mean? "$ langle M rangle_t $" or "$ langle M rangle_ infty $", where $ M_t $ is a continuous martingale

Suppose that $ M_t $ is a continuous martingale. I saw this notation in a newspaper: $ langle M rangle_t $ or $ langle M rangle_ infty $. What does this notation mean?

The paper does not explain the notation, so I suspect that it is known. I am aware that angle brackets are sometimes used to indicate expectation, but paper is used $ E M_t $ denote the expectation. Likewise, $ t $ is outside the parenthesis, which indicates that it is not an expectation notation. Does it refer to a particular notation used in martingales?

Probability – Is the assignment of the internal / external key figure continuous?

To let $ mathcal F $ be a field of subsets of a set $ Omega $. Equip the room $ (0.1) ^ mathcal F $ from functions $ mathcal F $ in $ (0.1) $ with the product topology. Then the set $ Delta $ from finally additive probability measures $ mathcal F $ is a convex and compact subset of $ (0.1) ^ mathcal F $.

If $ mu in Delta $, define the inner dimension $ mu ^ i: 2 ^ Omega to (0.1) $ to the $ mu $ by
$$ mu ^ i (A) = sup big { mu (F): F subset A, F in mathcal F big }, A subset Omega. $$
We can consider the inner dimension as a picture $ mu mapsto mu ^ i $ from $ Delta $ in $ (0.1) ^ {2 ^ Omega} $.

Question. Is the assignment of the internal key figure continuous? That is, if $ mu _ { alpha} $ is a network in $ Delta $ that converges to $ mu $ (i.e. $ mu_ alpha (F) to mu (F) $ for all $ F in mathcal F $) then it is true that $ mu ^ i_ alpha to mu ^ i $ (i.e. $ mu_ alpha ^ i (A) to mu ^ i (A) $ for each subset $ A $ from $ Omega $)?

A similar question arises for the external measure, although I assume that the answers are the same.

Performance – Olympus OM-D E-M1 Mark III: Is there a way to determine the number of exposures remaining in continuous shots before the buffer fills up?

Many interchangeable lens cameras offer some kind of buffer capacity indicator during recording, which typically shows the approximate number of images that can be stored in the buffer, and therefore how many more images can be captured in continuous mode before the camera slows down:

However, I cannot find a similar function on my Olympus OM-D E-M1 Mark III. Is there such a function in the camera? If so, where can I activate it? I've already reached Olympus also on Twittervia a display function for the buffer capacity.

Calculus – The inequality formula to prove $ x * 2 $ is continuous over a fixed finite interval [a, b]

This is discussed in Courant's introduction to Calculus and Analysis I. $ f (x) = x ^ 2 $ is continuous over a fixed finite interval (a, b).

It is written as:
$ | f (x) – f (x_ {0}) | = | x ^ 2 – x_ {0} ^ 2 | = | x – x_ {0} || x + x_ {0} | leqslant 2 | x – x_ {0} | (| b | + | a |) <2 sigma (| b | + | a |) <2 epsilon $

I have no idea what inequality sentences the author used:

  1. $ | x – x_ {0} || x + x_ {0} | leqslant 2 | x – x_ {0} | (| b | + | a |) $

  2. $ 2 | x – x_ {0} | (| b | + | a |) <2 sigma (| b | + | a |) $

What is continuous insulation? And why should you care? | Proxies-free

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Is there a SHARED light stand and softbox that would work with both flash lights and continuous LED light?

In the near future I would need continuous LED light for videography (on stage, like the bride and groom's platform at the Indian wedding).
I would also need studio flash units if I wanted to shoot models in the studio.
Therefore, I plan to buy both studio flashes (like Godox SK 400 II) and continuous LED light (like Simpex Pro HD LED).

My question now is: Is there a common light stand and common light modifiers (like softbox – rectangular or octagonal etc.) that can be used to set up flash and continuous light?
The light holders in the stand would probably differ and can be set on a case-by-case basis (flash or continuous)?

Reference requirement – dual space of continuous functions with Banach space value

To let $ X $ be a Banach room and $ K $ somewhat compact Hausdorff room. I am interested in the dual space of the Banach area
$$ C (K; X) = lbrace f: K to X, f text {is continuous} rbrace, qquad lVert f rVert_ infty: = max_ {x in K} lVert f (x) rVert. $$
In the event of, $ X = mathbb C $It is known that the dual space is given by the space of all regular Borel measures $ operatorname {rca} (K) $ in the room $ K $.

Now I think in the general case you can not hope that $ X $ is at least a Banach space, so that this characterization can be transferred to the vector-valued case, since a kind of lattice structure is required to define the regularity for vector-valued measures. Also, if you look at that $ L ^ p $-Situation, $ 1 leq p < infty $, one has $ L ^ p ( Omega; X) & # 39; = L ^ {p & # 39;} ( Omega; X & # 39;) $ exactly when the room $ X $ has the radon nicodymium property. A natural guess for me would be something like this:

If $ X $ is a Banach lattice, possibly with some additional Banach space or lattice properties, e.g. B. radon nicodymium or order continuity of the norm, then you have $ C (K; X) & # 39; = operatorname {rca} (K; X & # 39;) $, Where $ operatorname {rca} (K; X) $ denotes the space of $ X & # 39; $-evaluated regular Borel measures.

I would expect people to have already asked and solved this question. So my question is whether a result of this kind of taste is known? Are there any good references for this type of result? Thank you in advance!

Functional Analysis – Is the automorphism of the switch for $ C ^ * $ algebras with a continuous trace internal?

If $ R $ is a commutative ring, and $ A $ an azumaya algebra is over $ R $, then the switch (or flip or exchange etc.) automorphism of $ A otimes_R A $, given by $ a otimes b mapsto b otimes a $is internal: it is the conjugation by the so-called Goldman element $ g in (A otimes_R A) ^ times $, This element is an important feature of Azumaya algebras and can be defined as the element that corresponds to the reduced trace of $ A $ under the natural $ R $-Modulidentifikation $ A otimes_R A simeq operatorname {End} _R (A) $ (where the reduced trace is seen as a $ R $– linear map $ A to R subset A $).

Now in the context of $ C ^ * $-Algebras if $ R $ is commutative $ C ^ * $-Algebra (corresponding to some $ C_0 (T) $), my understanding is that the equivalent of Azumaya algebras is over $ R $ are the algebras with a continuous trace $ A $ with spectrum $ T $, For example, $ A $ is locally morita equivalent to $ R $;; it also has a Morita inverse: $ A otimes_T bar {A} $ is Morita equivalent to $ R $, Where $ otimes_T $ is the balanced tensor product relative to the spectrum $ T $,

The theory of continuous tracking $ C ^ * $ Algebras have many similarities to the theory of Azumaya algebras, but I couldn't tell if they share this property:

Is there a "Goldman element" $ g in (A otimes_T A) ^ times $ so that the conjugation through $ g $ is the switch automorphism? If so, can this element be chosen consistently?

Since then, the algebraic construction of the element can be imitated almost, but not exactly $ A otimes_T A $ can be identified as $ C_0 (T) $Module with $ mathcal {K} (Y_A) $ Where
$$ Y_A = {a in A , | , (t mapsto tr (aa ^ *) (t)) in C_0 (T) } $$
is a real Hilbert $ C_0 (T) $-Module. This is similar to $ A otimes_R A simeq operatorname {End} _R (A) $ in the algebraic case, however, there is no obvious "trace" element in $ mathcal {K} (Y_A) $,