wanyu/IteraTeR_full_sent huggingface dataset:
Original:  Using the recently available World Input-Output Database , we modeled the evolving world economic network ( from 1995 to 2011 ) with a series of time-homogeneous finite Markov chains.
Revised:  Using the recently available World Input-Output Database , we modeled the evolving world economic network ( from 1995 to 2011 through analysis of a series of time-homogeneous finite Markov chains.
Edit intent: clarity
Original:  Further, we introduced two measures of systemic risk, called systemic influence and systemic fragility, where the former is the ratio of number of influenced nodes to the total number of nodes, caused by a shock in the activity of that node and the latter is based on the number of times a specific economic node is affected by a shock in the activity of all the other nodes.
Revised:  Further, we introduced two measures of systemic risk, called systemic influence and systemic fragility, where the former is the ratio of number of influenced nodes to the total number of nodes, caused by a shock in the activity of a node and the latter is based on the number of times a specific economic node is affected by a shock in the activity of all the other nodes.
Edit intent: clarity
Original:  Further, we introduced two measures of systemic risk, called systemic influence and systemic fragility, where the former is the ratio of number of influenced nodes to the total number of nodes, caused by a shock in the activity of that node and the latter is based on the number of times a specific economic node is affected by a shock in the activity of all the other nodes.
Revised:  Further, we introduced two measures of systemic risk, called systemic influence and systemic fragility, where the former is the ratio of number of influenced nodes to the total number of nodes, caused by a shock in the activity of that node and the latter is based on the number of times a specific economic node is affected by a shock in the activity of any of the other nodes.
Edit intent: clarity
Original:  Finally, focusing on Kemeny constant as a global indicator of monetary flow across the network, we showed that there is a paradoxical effect of economic slow down of some of economic nodes on the overall flow of the world economic network.
Revised:  Finally, focusing on Kemeny constant as a global indicator of monetary flow across the network, we showed that there is a paradoxical effect of a change in activity levels of economic nodes on the overall flow of the world economic network.
Edit intent: clarity
Original:  Finally, focusing on Kemeny constant as a global indicator of monetary flow across the network, we showed that there is a paradoxical effect of economic slow down of some of economic nodes on the overall flow of the world economic network.
Revised:  Finally, focusing on Kemeny constant as a global indicator of monetary flow across the network, we showed that there is a paradoxical effect of economic slow down of some of economic nodes on the overall flow of the network.
Edit intent: clarity
Original:  While the economic slowdown of the majority of nodes with high structural power results to a slower average monetary flow  , there are some nodes, where their slowdowns improve the overall quality of the network in terms of connectivity and the average flow of money .
Revised:  While the economic slowdown of the majority of nodes with high structural power results to a slower average monetary flow over the network , there are some nodes, where their slowdowns improve the overall quality of the network in terms of connectivity and the average flow of money .
Edit intent: clarity
Original:  While the economic slowdown of the majority of nodes with high structural power results to a slower average monetary flow  , there are some nodes, where their slowdowns improve the overall quality of the network in terms of connectivity and the average flow of money .
Revised:  While the economic slowdown of the majority of nodes with high structural power results to a slower average monetary flow , there are some nodes, where their slowdowns improve the overall quality of the network in terms of connectivity and the average monetary flow .
Edit intent: clarity
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U %DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% \mathcal{F the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: meaning-changed
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded \mathcal{F semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: meaning-changed
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U %DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% \Omega to the reals and by \mathcal{L(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: meaning-changed
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| \mathcal{G )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U %DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: fluency
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% \mathcal{F )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: meaning-changed
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( \mathcal{G ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Edit intent: meaning-changed
Original:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
Revised:  Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( \mathcal{G ) invariant.
Edit intent: fluency
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  It is shown that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: clarity
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a \mathcal{G set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _{\mathcal{G} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities which are concentrated on atoms of \mathcal{G with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| \mathcal{G)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _{\mathcal{G ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( \omega )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: fluency
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} E_P[X] if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed
Original:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
Revised:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | \mathcal{G ) is pointwise continuous from below and continuous from above on the continuous functions.
Edit intent: meaning-changed